Section 26

Geometry of Euclidean space

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Subsection 26.1Orthogonal complement of a subspace (26.1-26.21)[21/67%]
Problem 26.1

Let aa be a nonzero vector of an nn-dimensional Euclidean space. Prove that the equation (a,x)=0(a, x) = 0 defines a subspace of dimension n−1n - 1.

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Problem 26.2

Let PP and QQ be sets of vectors of a Euclidean space such that (x,y)=0(x, y) = 0 for any x∈Px \in P and y∈Qy \in Q. Prove that the linear spans of these sets are orthogonal.

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Problem 26.3

In a Euclidean space E\mathcal{E} find the orthogonal complements of

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(1)

the zero subspace

(2)

the space E\mathcal{E}.

Problem 26.4

Let the subspaces L1,…,Ls\mathcal{L}_{1}, \ldots , \mathcal{L}_{s} of a Euclidean space be pairwise orthogonal. Prove that L1+…+Ls\mathcal{L}_{1} + \ldots + \mathcal{L}_{s} is a direct sum.

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Problem 26.5

Prove the following properties of the operation of passing to the orthogonal complement:

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(1)

(L1+L2)⊥=L1⊥∩L2⊥\left(\mathcal{L}_{1} + \mathcal{L}_{2}\right)^{\perp } = \mathcal{L}_{1}^{\perp } \cap \mathcal{L}_{2}^{\perp }

(2)

(L1∩L2)⊥=L1⊥+L2⊥\left(\mathcal{L}_{1} \cap \mathcal{L}_{2}\right)^{\perp } = \mathcal{L}_{1}^{\perp } + \mathcal{L}_{2}^{\perp }

(3)

(L⊥)⊥=L\left(\mathcal{L}^{\perp }\right)^{\perp } = \mathcal{L}.

Problem 26.6

The subspaces L1\mathcal{L}_{1} and L2\mathcal{L}_{2} are orthogonal. Are L1⊥\mathcal{L}_{1}^{\perp } and L2⊥\mathcal{L}_{2}^{\perp } necessarily orthogonal?

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Problem 26.7

Find a normalized vector orthogonal to the given ones:

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(1)

[404],[265]\left[\begin{smallmatrix} 4 \\ 0 \\ 4 \end{smallmatrix}\right], \left[\begin{smallmatrix} 2 \\ 6 \\ 5 \end{smallmatrix}\right], the basis is orthonormal

(2)

[2321],[1012],[0100]\left[\begin{smallmatrix} 2 \\ 3 \\ 2 \\ 1 \end{smallmatrix}\right], \left[\begin{smallmatrix} 1 \\ 0 \\ 1 \\ 2 \end{smallmatrix}\right], \left[\begin{smallmatrix} 0 \\ 1 \\ 0 \\ 0 \end{smallmatrix}\right], the basis is orthonormal

(3)

[31]\left[\begin{smallmatrix} 3 \\ 1 \end{smallmatrix}\right], the basis has Gram matrix A56\mathbf{A}_{56}. A56=[1113]\mathbf{A}_{56} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 3 \end{smallmatrix}\right].

(4)

[−110],[011]\left[\begin{smallmatrix} -1 \\ 1 \\ 0 \end{smallmatrix}\right], \left[\begin{smallmatrix} 0 \\ 1 \\ 1 \end{smallmatrix}\right], the basis has Gram matrix A207\mathbf{A}_{207}. A207=[12025−20−25]\mathbf{A}_{207} = \left[\begin{smallmatrix} 1 & 2 & 0 \\ 2 & 5 & -2 \\ 0 & -2 & 5 \end{smallmatrix}\right].

Problem 26.8

A subspace L\mathcal{L} is given in an orthonormal basis by a system of linear equations Aξ=oA\xi = o. Find:

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(1)

a basis in L⊥\mathcal{L}^{\perp }

(2)

a system of equations of the subspace L⊥\mathcal{L}^{\perp }.

Problem 26.9

Let a1,…,aka_{1}, \ldots , a_{k} be a basis of the subspace L\mathcal{L}, and let the coordinate columns of the vectors a1,…,aka_{1}, \ldots , a_{k} in an orthonormal basis of the space E\mathcal{E} form the matrix AA. Find:

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(1)

a basis in L⊥\mathcal{L}^{\perp }

(2)

a system of equations of the subspace L⊥\mathcal{L}^{\perp }.

Problem 26.10

Solve problem 18. 20 using geometric considerations.

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Problem 26.11

A subspace L\mathcal{L} is given in a basis ee with Gram matrix Γ\Gamma by a system of linear equations Aξ=oA\xi = \mathbf{o}. Find:

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(1)

a basis in L⊥\mathcal{L}^{\perp }

(2)

the matrix of the system of equations of the subspace L⊥\mathcal{L}^{\perp }.

Problem 26.12

Let a1,…,aka_{1}, \ldots , a_{k} be a basis of the subspace L\mathcal{L}, and let the coordinate columns of the vectors a1,…,aka_{1}, \ldots , a_{k} in a basis ee of the space E\mathcal{E} form the matrix AA. The Gram matrix Γ\Gamma of the basis e is given. Find:

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(1)

a basis in L⊥\mathcal{L}^{\perp }

(2)

the matrix of the system of equations of the subspace L⊥\mathcal{L}^{\perp }.

Problem 26.13

The subspace L\mathcal{L} is given as the linear span of vectors having, in an orthonormal basis, the coordinate columns:

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(1)

[312]\left[\begin{smallmatrix} 3 \\ 1 \\ 2\end{smallmatrix}\right];

(2)

[1−51]\left[\begin{smallmatrix} 1 \\ -5 \\ 1\end{smallmatrix}\right], [−111]\left[\begin{smallmatrix} -1 \\ 1 \\ 1\end{smallmatrix}\right];

(3)

[3−1591]\left[\begin{smallmatrix} 3 \\ -15 \\ 9 \\ 1\end{smallmatrix}\right], [3−6−32]\left[\begin{smallmatrix} 3 \\ - 6 \\ - 3 \\ 2\end{smallmatrix}\right];

(4)

[43−32]\left[\begin{smallmatrix} 4 \\ 3 \\ -3 \\ 2\end{smallmatrix}\right], [−132−3]\left[\begin{smallmatrix} -1 \\ 3 \\ 2 \\ -3\end{smallmatrix}\right], [291−4]\left[\begin{smallmatrix} 2 \\ 9 \\ 1 \\ -4\end{smallmatrix}\right].

Find:

(1) the matrix of the system of equations defining L⊥\mathcal{L}^{\perp },

(2) a basis in L⊥\mathcal{L}^{\perp }.

Problem 26.14

A subspace L\mathcal{L} is given in an orthonormal basis by a system of linear equations Aξ=oA\xi = \mathbf{o}. Find a basis of the subspace L⊥\mathcal{L}^{\perp } if the matrix AA equals:

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(1)

[321]\left[\begin{smallmatrix} 3 & 2 & 1 \end{smallmatrix}\right]

(2)

[1−51−111]\left[\begin{smallmatrix} 1 & - 5 & 1 \\ - 1 & 1 & 1\end{smallmatrix}\right]

(3)

[13−122−135110−61]\left[\begin{smallmatrix} 1 & 3 & - 1 & 2 \\ 2 & - 1 & 3 & 5 \\ 1 & 10 & - 6 & 1\end{smallmatrix}\right]

(4)

[1−5−61151−4318715]\left[\begin{smallmatrix} 1 & - 5 & - 6 & 11 \\ 5 & 1 & - 4 & 3 \\ 1 & 8 & 7 & 15\end{smallmatrix}\right]

(5)

A519\mathbf{A}_{519}. A519=[12313211231−122215520]\mathbf{A}_{519} = \left[\begin{smallmatrix} 1 & 2 & 3 & 1 \\ 3 & 2 & 1 & 1 \\ 2 & 3 & 1 & -1 \\ 2 & 2 & 2 & 1 \\ 5 & 5 & 2 & 0 \end{smallmatrix}\right].

(6)

A582\mathbf{A}_{582}. A582=[33−3−341−111−221−1−11]\mathbf{A}_{582} = \left[\begin{smallmatrix} 3 & 3 & -3 & -3 & 4 \\ 1 & -1 & 1 & 1 & -2 \\ 2 & 1 & -1 & -1 & 1 \end{smallmatrix}\right].

(7)

A518\mathbf{A}_{518}. A518=[0−22−816824−1604−416]\mathbf{A}_{518} = \left[\begin{smallmatrix} 0 & -2 & 2 & -8 \\ 16 & 8 & 24 & -16 \\ 0 & 4 & -4 & 16 \end{smallmatrix}\right].

(8)

A514\mathbf{A}_{514}. A514=[2−3574−6232−3−11−15]\mathbf{A}_{514} = \left[\begin{smallmatrix} 2 & -3 & 5 & 7 \\ 4 & -6 & 2 & 3 \\ 2 & -3 & -11 & -15 \end{smallmatrix}\right].

Problem 26.15

A subspace L\mathcal{L} is given in an orthonormal basis by a system of linear equations Aξ=oA\xi = \mathbf{o}. Find a system of equations of the subspace L⊥\mathcal{L}^{\perp }:

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(1)

x1−x2+x3+x4=08x1−x2+2x3+4x4=011x1+2x2−x3−x4=0\begin{aligned} & x_{1} - x_{2} + x_{3} + x_{4} = 0 \\ & 8x_{1} - x_{2} + 2x_{3} + 4x_{4} = 0 \\ & 11x_{1} + 2x_{2} - x_{3} - x_{4} = 0\end{aligned}

(2)

8x1−x2+2x3−4x4=0−6x1+3x2−4x3+6x4=0\begin{aligned} & 8x_{1} - x_{2} + 2x_{3} - 4x_{4} = 0 \\ & - 6x_{1} + 3x_{2} - 4x_{3} + 6x_{4} = 0\end{aligned}

(3)

3x1+5x2+x3+3x4+11x5=04x1+7x2+2x3+5x4+16x5=0\begin{aligned} & 3x_{1} + 5x_{2} + x_{3} + 3x_{4} + 11x_{5} = 0 \\ & 4x_{1} + 7x_{2} + 2x_{3} + 5x_{4} + 16x_{5} = 0\end{aligned}

(4)

5x1+24x2−7x3−3x4=0−x1−2x2+7x3+3x4=0\begin{aligned} 5x_{1} + 24x_{2} - 7x_{3} - 3x_{4} & = 0 \\ - x_{1} - 2x_{2} + 7x_{3} + 3x_{4} & = 0\end{aligned}

(5)

The system of equations has matrix

(1) A573\mathbf{A}_{573},

(2) A583\mathbf{A}_{583}. A573=[0011101001],A583=[−2111−11−12−117−51−55]\mathbf{A}_{573} = \left[\begin{smallmatrix} 0 & 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 & 1 \end{smallmatrix}\right], \mathbf{A}_{583} = \left[\begin{smallmatrix} -2 & 1 & 1 & 1 & -1 \\ 1 & -1 & 2 & -1 & 1 \\ 7 & -5 & 1 & -5 & 5 \end{smallmatrix}\right].

Problem 26.16

Find:

(1) a basis in L⊥\mathcal{L}^{\perp };

(2) the matrix of the system of equations of the subspace L⊥\mathcal{L}^{\perp }, if the subspace L\mathcal{L} is given in a basis e\mathbf{e} with Gram matrix Γ\Gamma by a system of linear equations Aξ=oA\xi = \mathbf{o}:

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(1)

A=[121],Γ=A207A =\left[\begin{smallmatrix} 1 \\ 2 \\ 1\end{smallmatrix}\right], \Gamma = \mathbf{A}_{207}. A207=[12025−20−25]\mathbf{A}_{207} = \left[\begin{smallmatrix} 1 & 2 & 0 \\ 2 & 5 & -2 \\ 0 & -2 & 5 \end{smallmatrix}\right].

(2)

A=[101110],Γ=A385A =\left[\begin{smallmatrix} 1 & 0 & 1 \\ 1 & 1 & 0\end{smallmatrix}\right], \Gamma = \mathbf{A}_{385}. A385=[1−23−25−83−814]\mathbf{A}_{385} = \left[\begin{smallmatrix} 1 & -2 & 3 \\ -2 & 5 & -8 \\ 3 & -8 & 14 \end{smallmatrix}\right].

(3)

A=[312111],Γ=A386A =\left[\begin{smallmatrix} 3 & 1 & 2 \\ 1 & 1 & 1\end{smallmatrix}\right], \Gamma = \mathbf{A}_{386}. A386=[20−1021−112]\mathbf{A}_{386} = \left[\begin{smallmatrix} 2 & 0 & -1 \\ 0 & 2 & 1 \\ -1 & 1 & 2 \end{smallmatrix}\right].

(4)

A=[−11011−210],Γ=A424A =\left[\begin{smallmatrix} - 1 & 1 & 0 & 1 \\ 1 & - 2 & 1 & 0\end{smallmatrix}\right], \Gamma = \mathbf{A}_{424}. A424=[200−102−100−120−1002]\mathbf{A}_{424} = \left[\begin{smallmatrix} 2 & 0 & 0 & -1 \\ 0 & 2 & -1 & 0 \\ 0 & -1 & 2 & 0 \\ -1 & 0 & 0 & 2 \end{smallmatrix}\right].

Problem 26.17

Find

(1) a basis in L⊥\mathcal{L}^{\perp };

(2) a system of equations of the subspace L⊥\mathcal{L}^{\perp }, if a1,…,aka_{1}, \ldots , a_{k} is a basis of the subspace L\mathcal{L}, and the coordinate columns of the vectors a1,…,aka_{1}, \ldots , a_{k} in a basis ee of the space E\mathcal{E} form the matrix AA. The Gram matrix Γ\Gamma of the basis e is also given:

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(1)

A=[112],Γ=A207A =\left[\begin{smallmatrix} 1 \\ 1 \\ 2\end{smallmatrix}\right], \quad \Gamma = \mathbf{A}_{207}. A207=[12025−20−25]\mathbf{A}_{207} = \left[\begin{smallmatrix} 1 & 2 & 0 \\ 2 & 5 & -2 \\ 0 & -2 & 5 \end{smallmatrix}\right].

(2)

A=[112011]A =\left[\begin{smallmatrix} 1 & 1 \\ 2 & 0 \\ 1 & 1\end{smallmatrix}\right], Γ=A385\Gamma = \mathbf{A}_{385}. A385=[1−23−25−83−814]\mathbf{A}_{385} = \left[\begin{smallmatrix} 1 & -2 & 3 \\ -2 & 5 & -8 \\ 3 & -8 & 14 \end{smallmatrix}\right].

(3)

A=[111112],Γ=A386A =\left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \\ 1 & 2\end{smallmatrix}\right], \quad \Gamma = \mathbf{A}_{386}. A386=[20−1021−112]\mathbf{A}_{386} = \left[\begin{smallmatrix} 2 & 0 & -1 \\ 0 & 2 & 1 \\ -1 & 1 & 2 \end{smallmatrix}\right].

(4)

A=[12211122],Γ=A424A =\left[\begin{smallmatrix} 1 & 2 \\ 2 & 1 \\ 1 & 1 \\ 2 & 2\end{smallmatrix}\right], \quad \Gamma = \mathbf{A}_{424}. A424=[200−102−100−120−1002]\mathbf{A}_{424} = \left[\begin{smallmatrix} 2 & 0 & 0 & -1 \\ 0 & 2 & -1 & 0 \\ 0 & -1 & 2 & 0 \\ -1 & 0 & 0 & 2 \end{smallmatrix}\right].

Problem 26.18

In the space of square matrices with the standard scalar product, find the orthogonal complement of the subspace of

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(1)

matrices with trace equal to zero

(2)

upper triangular matrices.

Problem 26.19

In the space of polynomials of degree at most nn with the standard scalar product, find the orthogonal complement of the subspace of polynomials of even degree.

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Problem 26.20

Let a Euclidean space E\mathcal{E} be the direct sum of subspaces Li(i=1,…,s)\mathcal{L}_{i}(i = 1, \ldots , s), and x=∑xix = \sum x_{i}, y=∑yiy = \sum y_{i} (xi,yi∈Li)\left(x_{i}, y_{i} \in \mathcal{L}_{i}\right). Prove that the subspaces Li\mathcal{L}_{i} are pairwise orthogonal if (x,y)=∑(xi,yi)(x, y) = \sum \left(x_{i}, y_{i}\right) for any xx and yy.

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Problem 26.21
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(1)

To find the coefficients of the expansion of a vector b⃗\vec{b} in terms of the vectors a1→\overrightarrow {a_{1}} and a2→\overrightarrow {a_{2}}, a system of three linear equations in two unknowns is formed. Establish that the Fredholm theorem for this system is equivalent to the following (geometrically obvious) statement: the vector b⃗\vec{b} expands in terms of a1→\overrightarrow {a_{1}} and a2→\overrightarrow {a_{2}} if and only if it is orthogonal to every vector y⃗\vec{y} orthogonal to these vectors.

(2)

Prove the Fredholm theorem using the result of problem 26.5,3)26. 5, 3).

Subsection 26.2Orthogonal projections (26.22-26.41)[20/70%]
Problem 26.22

In a Euclidean space E\mathcal{E} a vector xx is given. Find its orthogonal projection and orthogonal component when projecting

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(1)

onto the zero subspace;

(2)

onto E\mathcal{E}.

Problem 26.23

In the subspace L⊂E\mathcal{L} \subset \mathcal{E} a basis a1,…,aka_{1}, \ldots , a_{k} is given. In an orthonormal basis of the space E\mathcal{E} the coordinate columns of these vectors form the matrix AA. A vector x∈Ex\in \mathcal{E} is given by its coordinate column ξ\xi. Find the orthogonal projections x′x^{\prime } and x′′x^{\prime \prime } of the vector xx onto L\mathcal{L} and L⊥\mathcal{L}^{\perp }.

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Problem 26.24

In an orthonormal basis, a subspace L\mathcal{L} is given by a system of linear equations with matrix AA (the rows of AA are linearly independent). A vector xx is given by its coordinate column ξ\xi. Find the orthogonal projections x′x^{\prime } and x′′x^{\prime \prime } of the vector xx onto L\mathcal{L} and L⊥\mathcal{L}^{\perp }.

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Problem 26.25

In the space E\mathcal{E} a basis ee with Gram matrix Γ\Gamma is chosen. The subspace L\mathcal{L} is spanned by linearly independent vectors a1,…,aka_{1}, \ldots , a_{k}, whose coordinate columns form the matrix AA. A vector x∈Ex\in \mathcal{E} is given by its coordinate column ξ\xi. Find the orthogonal projections x′x^{\prime } and x′′x^{\prime \prime } of the vector xx onto L\mathcal{L} and L⊥\mathcal{L}^{\perp }.

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Problem 26.26

In a basis ee with Gram matrix Γ\Gamma, a subspace L\mathcal{L} is given by a system of linear equations with matrix AA (the rows of AA are linearly independent). A vector xx is given by its coordinate column ξ\xi. Find the orthogonal projections x′x^{\prime } and x′′x^{\prime \prime } of the vector xx onto L\mathcal{L} and L⊥\mathcal{L}^{\perp }.

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Problem 26.27

The subspace L\mathcal{L} is the linear span of vectors a1,…,aka_{1}, \ldots , a_{k}. In an orthonormal basis the coordinate columns of these vectors and the coordinate column ξ\xi of the vector xx are given. Find the coordinate columns ξ′\xi^{\prime } and ξ′′\xi^{\prime \prime } of the orthogonal projections of the vector xx onto L\mathcal{L} and L⊥\mathcal{L}^{\perp }, respectively:

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(1)

a1=[1055],ξ=[300]\mathbf{a}_{1} =\left[\begin{smallmatrix} 10 \\ 5 \\ 5\end{smallmatrix}\right], \quad \xi =\left[\begin{smallmatrix} 3 \\ 0 \\ 0\end{smallmatrix}\right]

(2)

a1=[615],a2=[4−13],ξ=[13−2]\mathbf{a}_{1} =\left[\begin{smallmatrix} 6 \\ 1 \\ 5\end{smallmatrix}\right], \quad \mathbf{a}_{2} =\left[\begin{smallmatrix} 4 \\ - 1 \\ 3\end{smallmatrix}\right], \quad \xi =\left[\begin{smallmatrix} 1 \\ 3 \\ - 2\end{smallmatrix}\right]

(3)

a1=[2112],ξ=[5370]\mathbf{a}_{1} =\left[\begin{smallmatrix} 2 \\ 1 \\ 1 \\ 2\end{smallmatrix}\right], \quad \xi =\left[\begin{smallmatrix} 5 \\ 3 \\ 7 \\ 0\end{smallmatrix}\right]

(4)

a1=[3−211],a2=[10−11]\mathbf{a}_{1} =\left[\begin{smallmatrix} 3 \\ - 2 \\ 1 \\ 1\end{smallmatrix}\right], \mathbf{a}_{2} =\left[\begin{smallmatrix} 1 \\ 0 \\ - 1 \\ 1\end{smallmatrix}\right], ξ=[2−13−2]\xi =\left[\begin{smallmatrix} 2 \\ - 1 \\ 3 \\ - 2\end{smallmatrix}\right]

(5)

a1=[2301],a2=[05−2−1],ξ=[6042]\mathbf{a}_{1} =\left[\begin{smallmatrix} 2 \\ 3 \\ 0 \\ 1\end{smallmatrix}\right], \mathbf{a}_{2} =\left[\begin{smallmatrix} 0 \\ 5 \\ - 2 \\ - 1\end{smallmatrix}\right], \xi =\left[\begin{smallmatrix} 6 \\ 0 \\ 4 \\ 2\end{smallmatrix}\right]

(6)

a1=[43−32],a2=[291−4]\mathbf{a}_{1} =\left[\begin{smallmatrix} 4 \\ 3 \\ - 3 \\ 2\end{smallmatrix}\right], \quad \mathbf{a}_{2} =\left[\begin{smallmatrix} 2 \\ 9 \\ 1 \\ - 4\end{smallmatrix}\right], a3=[−132−3],ξ=[24−13]\mathbf{a}_{3} =\left[\begin{smallmatrix} - 1 \\ 3 \\ 2 \\ - 3\end{smallmatrix}\right], \quad \xi =\left[\begin{smallmatrix} 2 \\ 4 \\ - 1 \\ 3\end{smallmatrix}\right]

(7)

a1=[31−2−2],a2=[31−1−3]\mathbf{a}_{1} =\left[\begin{smallmatrix} 3 \\ 1 \\ - 2 \\ - 2\end{smallmatrix}\right], \quad \mathbf{a}_{2} =\left[\begin{smallmatrix} 3 \\ 1 \\ - 1 \\ - 3\end{smallmatrix}\right], a3=[3−10−2],ξ=[64−2−4]\mathbf{a}_{3} =\left[\begin{smallmatrix} 3 \\ - 1 \\ 0 \\ - 2\end{smallmatrix}\right], \quad \xi =\left[\begin{smallmatrix} 6 \\ 4 \\ - 2 \\ - 4\end{smallmatrix}\right]

(8)

a1=[3112],a2=[4021],a3=[8023]\mathbf{a}_{1} =\left[\begin{smallmatrix} 3 \\ 1 \\ 1 \\ 2\end{smallmatrix}\right], \quad \mathbf{a}_{2} =\left[\begin{smallmatrix} 4 \\ 0 \\ 2 \\ 1\end{smallmatrix}\right], \mathbf{a}_{3} =\left[\begin{smallmatrix} 8 \\ 0 \\ 2 \\ 3\end{smallmatrix}\right], ξ=[−7131]\xi =\left[\begin{smallmatrix} - 7 \\ 1 \\ 3 \\ 1\end{smallmatrix}\right]

(9)

a1=c201,a2=c202,ξ=c168\mathbf{a}_{1} = \mathbf{c}_{201}, \quad \mathbf{a}_{2} = \mathbf{c}_{202}, \quad \xi = \mathbf{c}_{168}. c201=[1−75−2],c202=[1013],c168=[0100]\mathbf{c}_{201} = \left[\begin{smallmatrix} 1 \\ -7 \\ 5 \\ -2 \end{smallmatrix}\right], \mathbf{c}_{202} = \left[\begin{smallmatrix} 1 \\ 0 \\ 1 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{168} = \left[\begin{smallmatrix} 0 \\ 1 \\ 0 \\ 0 \end{smallmatrix}\right].

(10)

a1=c176,a2=c161,ξ=c193\mathbf{a}_{1} = \mathbf{c}_{176}, \quad \mathbf{a}_{2} = \mathbf{c}_{161}, \quad \xi = \mathbf{c}_{193}. c176=[−11−2−1],c161=[1301],c193=[0001]\mathbf{c}_{176} = \left[\begin{smallmatrix} -1 \\ 1 \\ -2 \\ -1 \end{smallmatrix}\right], \mathbf{c}_{161} = \left[\begin{smallmatrix} 1 \\ 3 \\ 0 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{193} = \left[\begin{smallmatrix} 0 \\ 0 \\ 0 \\ 1 \end{smallmatrix}\right].

(11)

a1=c236,a2=c271,a3=c283,ξ=c232\mathbf{a}_{1} = \mathbf{c}_{236}, \mathbf{a}_{2} = \mathbf{c}_{271}, \quad \mathbf{a}_{3} = \mathbf{c}_{283}, \quad \xi = \mathbf{c}_{232}. c236=[−2−1012],c271=[1−12−11],c283=[1−2−3−21],c232=[220−2−2]\mathbf{c}_{236} = \left[\begin{smallmatrix} -2 \\ -1 \\ 0 \\ 1 \\ 2 \end{smallmatrix}\right], \mathbf{c}_{271} = \left[\begin{smallmatrix} 1 \\ -1 \\ 2 \\ -1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{283} = \left[\begin{smallmatrix} 1 \\ -2 \\ -3 \\ -2 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{232} = \left[\begin{smallmatrix} 2 \\ 2 \\ 0 \\ -2 \\ -2 \end{smallmatrix}\right].

Problem 26.28

In an orthonormal basis, a subspace L\mathcal{L} is given by a system of linear equations with matrix AA, and a vector xx -- by its coordinate column ξ\xi. Find the projections of xx onto L\mathcal{L} and onto L⊥:\mathcal{L}^{\perp }:

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(1)

A=[111],ξ=[123]A =\left[\begin{smallmatrix} 1 & 1 & 1\end{smallmatrix}\right], \quad \xi =\left[\begin{smallmatrix} 1 \\ 2 \\ 3\end{smallmatrix}\right]

(2)

A=[32−5−1−12],ξ=[426]A =\left[\begin{smallmatrix} 3 & 2 & - 5 \\ - 1 & - 1 & 2\end{smallmatrix}\right], \quad \xi =\left[\begin{smallmatrix} 4 \\ 2 \\ 6\end{smallmatrix}\right]

(3)

A=[11111],ξ=[54321]A =\left[\begin{smallmatrix} 1 & 1 & 1 & 1 & 1\end{smallmatrix}\right], \quad \xi =\left[\begin{smallmatrix} 5 \\ 4 \\ 3 \\ 2 \\ 1\end{smallmatrix}\right]

(4)

A=[2−3513−561],ξ=[7−594]A =\left[\begin{smallmatrix} 2 & - 3 & 5 & 1 \\ 3 & - 5 & 6 & 1\end{smallmatrix}\right], \quad \xi =\left[\begin{smallmatrix} 7 \\ - 5 \\ 9 \\ 4\end{smallmatrix}\right]

(5)

A=[435−2324−1],ξ=[−2420]A =\left[\begin{smallmatrix} 4 & 3 & 5 & -2 \\ 3 & 2 & 4 & - 1\end{smallmatrix}\right], \quad \xi =\left[\begin{smallmatrix} -2 \\ 4 \\ 2 \\ 0\end{smallmatrix}\right]

(6)

A=[1031341018311],ξ=[8−53−1]A =\left[\begin{smallmatrix} 10 & 3 & 1 & 3 \\ 4 & 1 & 0 & 1 \\ 8 & 3 & 1 & 1\end{smallmatrix}\right], \quad \xi =\left[\begin{smallmatrix} 8 \\ - 5 \\ 3 \\ - 1\end{smallmatrix}\right].

Problem 26.29

For the vectors and subspaces given in problem 26. 28:

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(1)

A=[111],ξ=[123]A =\left[\begin{smallmatrix} 1 & 1 & 1\end{smallmatrix}\right], \quad \xi =\left[\begin{smallmatrix} 1 \\ 2 \\ 3\end{smallmatrix}\right]

(2)

A=[32−5−1−12],ξ=[426]A =\left[\begin{smallmatrix} 3 & 2 & - 5 \\ - 1 & - 1 & 2\end{smallmatrix}\right], \quad \xi =\left[\begin{smallmatrix} 4 \\ 2 \\ 6\end{smallmatrix}\right]

(3)

A=[11111],ξ=[54321]A =\left[\begin{smallmatrix} 1 & 1 & 1 & 1 & 1\end{smallmatrix}\right], \quad \xi =\left[\begin{smallmatrix} 5 \\ 4 \\ 3 \\ 2 \\ 1\end{smallmatrix}\right]

(4)

A=[2−3513−561],ξ=[7−594]A =\left[\begin{smallmatrix} 2 & - 3 & 5 & 1 \\ 3 & - 5 & 6 & 1\end{smallmatrix}\right], \quad \xi =\left[\begin{smallmatrix} 7 \\ - 5 \\ 9 \\ 4\end{smallmatrix}\right]

(5)

A=[435−2324−1],ξ=[−2420]A =\left[\begin{smallmatrix} 4 & 3 & 5 & -2 \\ 3 & 2 & 4 & - 1\end{smallmatrix}\right], \quad \xi =\left[\begin{smallmatrix} -2 \\ 4 \\ 2 \\ 0\end{smallmatrix}\right]

(6)

A=[1031341018311],ξ=[8−53−1]A =\left[\begin{smallmatrix} 10 & 3 & 1 & 3 \\ 4 & 1 & 0 & 1 \\ 8 & 3 & 1 & 1\end{smallmatrix}\right], \quad \xi =\left[\begin{smallmatrix} 8 \\ - 5 \\ 3 \\ - 1\end{smallmatrix}\right].

find the coordinates of the vector yy obtained by reflecting the vector xx in the subspace L\mathcal{L}.

Problem 26.30

Find the orthogonal projection of the polynomial 35t4+15t3−15t2−8t+435t^{4} + 15t^{3} - 15t^{2} - 8t + 4 onto the subspace of polynomials of degree at most 2 in the space of polynomials:

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(1)

with the standard scalar product

(2)

with the scalar product defined in problem 25. 8(3).

Problem 26.31

In the space of polynomials with the standard scalar product, find the distance from the polynomial tnt^{n} to the linear span of the polynomials 1,t,…,tn−11, t, \ldots , t^{n - 1}:

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(1)

for n=2n = 2

(2)

for n=3n = 3.

Problem 26.32

Let f1,…,fkf_{1}, \ldots , f_{k} be a basis of the subspace L\mathcal{L}. Prove that the orthogonal projection of an arbitrary vector xx onto L\mathcal{L} equals the sum of its projections onto the one-dimensional subspaces spanned by f1,…,fkf_{1}, \ldots , f_{k} if and only if the basis is orthogonal.

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Problem 26.33

Let it be that for any vector x∈Ex\in \mathcal{E} the sum of its orthogonal projections onto the subspaces L1,L2⊂E\mathcal{L}_{1}, \mathcal{L}_{2} \subset \mathcal{E} equals the orthogonal projection of xx onto their sum L1+L2\mathcal{L}_{1} + \mathcal{L}_{2}. Prove that the subspaces L1\mathcal{L}_{1} and L2\mathcal{L}_{2} are orthogonal.

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Problem 26.34

Consider the space of functions continuous on the interval [−1,1]\left[ - 1, 1\right], with the scalar product defined in problem 25. 7. Expand the orthogonal projection of the function ff onto the subspace Pk\mathcal{P}_{k} of polynomials of degree at most kk in the basis consisting of the Legendre polynomials Pi(t)P_{i}(t) (problem 25. 9).

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Problem 26.35

Let x′x^{\prime } be the orthogonal projection of xx onto a subspace. Prove that ∣x′∣⩽∣x∣\left|x^{\prime }\right| \leqslant \left|x\right|, with equality holding if and only if x′=xx^{\prime } = x

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Problem 26.36

Let x′x^{\prime } be the orthogonal projection of xx onto the subspace L⊂E\mathcal{L} \subset \mathcal{E}, and let x′′x^{\prime \prime } be the orthogonal component. Prove that for any vector y∈Ly\in \mathcal{L} other than x′x^{\prime }, we have ∣x′′∣<∣x−y∣\left|x^{\prime \prime }\right| < \left|x - y\right|

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Problem 26.37

Let Li,(i=1,…,s)\mathcal{L}_{i}, \quad (i = 1, \ldots , s) be pairwise orthogonal subspaces of a Euclidean space. Prove that the sum of the squares of the lengths of the projections of an arbitrary vector xx onto these subspaces does not exceed ∣x∣2\left|x\right|^{2}, and that this bound is attained if xx belongs to the sum of the subspaces.

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Problem 26.38

Consider two subspaces L1⊆L2\mathcal{L}_{1} \subseteq \mathcal{L}_{2}. Denote by L′\mathcal{L}^{\prime } the orthogonal complement of L1\mathcal{L}_{1} in L2\mathcal{L}_{2}, and let x1,x2x_{1}, x_{2} and x′x^{\prime } be the orthogonal projections of the vector xx onto the subspaces L1,L2\mathcal{L}_{1}, \mathcal{L}_{2} and L′\mathcal{L}^{\prime }. Prove that:

?
(1)

x2=x1+x′x_{2} = x_{1} + x^{\prime }

(2)

∣x1∣⩽∣x2∣\left|x_{1}\right| \leqslant \left|x_{2}\right|, and for any xx equality holds if and only if L1=L2\mathcal{L}_{1} = \mathcal{L}_{2}.

Problem 26.39

Let L−k\mathcal{L} - k-dimensional subspace of an nn-dimensional Euclidean space E\mathcal{E}. Let also e1,…,ene_{1}, \ldots , e_{n} be an orthonormal basis in E\mathcal{E}, and let e1′,…,en′e_{1}^{\prime }, \ldots , e_{n}^{\prime } be the orthogonal projections of these vectors onto L\mathcal{L}. Prove that ∑i=1k∣ei′∣2=k\sum_{i = 1}^{k} \left|e_{i}^{\prime }\right|^{2} = k.

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Problem 26.40

Let e1,…,ene_{1}, \ldots , e_{n} be an orthonormal basis in a Euclidean space and let a system of vectors g1,…,gng_{1}, \ldots , g_{n} be such that ∑i=1n∣ei−gi∣<1\sum_{i = 1}^{n}\left|e_{i} - g_{i}\right| < 1. Prove that this system of vectors is linearly independent.

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Problem 26.41

Let e1,…,ene_{1}, \ldots , e_{n} be an orthonormal basis in a Euclidean space and let a system of vectors g1,…,gng_{1}, \ldots , g_{n} be such that ∑i=1ncos⁡(e^i,gi)⩾(2n−1)/2\sum_{i = 1}^{n} \cos \left(\widehat{e}_{i}, g_{i}\right) \geqslant (2n - 1) / 2. Prove that this system of vectors is linearly independent.

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Subsection 26.3Orthogonalization (26.42-26.48)[7/86%]
Problem 26.42

Orthogonalize the following systems of vectors of an arithmetic space with the standard scalar product:

?
(1)

[13−2]\left[\begin{smallmatrix} 1 \\ 3 \\ -2\end{smallmatrix}\right], [37−2]\left[\begin{smallmatrix} 3 \\ 7 \\ -2\end{smallmatrix}\right];

(2)

[210−1],[3626]\left[\begin{smallmatrix} 2 \\ 1 \\ 0 \\ - 1\end{smallmatrix}\right],\left[\begin{smallmatrix} 3 \\ 6 \\ 2 \\ 6\end{smallmatrix}\right]

(3)

[131],[513],[16−8]\left[\begin{smallmatrix} 1 \\ 3 \\ 1\end{smallmatrix}\right],\left[\begin{smallmatrix} 5 \\ 1 \\ 3\end{smallmatrix}\right],\left[\begin{smallmatrix} 1 \\ 6 \\ - 8\end{smallmatrix}\right]

(4)

[212],[622],[14−3]\left[\begin{smallmatrix} 2 \\ 1 \\ 2\end{smallmatrix}\right],\left[\begin{smallmatrix} 6 \\ 2 \\ 2\end{smallmatrix}\right], \quad \left[\begin{smallmatrix} 1 \\ 4 \\ - 3\end{smallmatrix}\right]

(5)

[123],[211],[6−7−2]\left[\begin{smallmatrix} 1 \\ 2 \\ 3\end{smallmatrix}\right],\left[\begin{smallmatrix} 2 \\ 1 \\ 1\end{smallmatrix}\right],\left[\begin{smallmatrix} 6 \\ -7 \\ - 2\end{smallmatrix}\right]

(6)

[1212]\left[\begin{smallmatrix} 1 \\ 2 \\ 1 \\ 2\end{smallmatrix}\right], [4041]\left[\begin{smallmatrix} 4 \\ 0 \\ 4 \\ 1\end{smallmatrix}\right], [113−1−3]\left[\begin{smallmatrix} 1 \\ 13 \\ -1 \\ -3\end{smallmatrix}\right]

(7)

[1−1−11]\left[\begin{smallmatrix} 1 \\ - 1 \\ - 1 \\ 1\end{smallmatrix}\right], [2332]\left[\begin{smallmatrix} 2 \\ 3 \\ 3 \\ 2\end{smallmatrix}\right], [4402]\left[\begin{smallmatrix} 4 \\ 4 \\ 0 \\ 2\end{smallmatrix}\right], [1−5−5−1]\left[\begin{smallmatrix} 1 \\ -5 \\ -5 \\ -1\end{smallmatrix}\right]

Problem 26.43

In a Euclidean space E\mathcal{E} an orthonormal basis e is chosen. In it are given the coordinates of the vectors of a basis a of the subspace L⊆E\mathcal{L} \subseteq \mathcal{E}. Using the orthogonalization process, find in ee the coordinate columns of the vectors of an orthonormal basis in L\mathcal{L} and write out the transition matrix from the basis a to this basis:

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(1)

[31],[9−7]\left[\begin{smallmatrix} 3 \\ 1\end{smallmatrix}\right],\left[\begin{smallmatrix} 9 \\ -7\end{smallmatrix}\right]

(2)

[110],[20−1],[003]\left[\begin{smallmatrix} 1 \\ 1 \\ 0\end{smallmatrix}\right],\left[\begin{smallmatrix} 2 \\ 0 \\ - 1\end{smallmatrix}\right],\left[\begin{smallmatrix} 0 \\ 0 \\ 3\end{smallmatrix}\right]

(3)

[11−22]\left[\begin{smallmatrix} 1 \\ 1 \\ -2\sqrt{2}\end{smallmatrix}\right], [3−1−2]\left[\begin{smallmatrix} 3 \\ -1 \\ -\sqrt{2}\end{smallmatrix}\right], [42−2]\left[\begin{smallmatrix} 4 \\ 2 \\ -\sqrt{2}\end{smallmatrix}\right]

(4)

[111−1]\left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ - 1\end{smallmatrix}\right], [331−1]\left[\begin{smallmatrix} 3 \\ 3 \\ 1 \\ - 1\end{smallmatrix}\right], [31−13]\left[\begin{smallmatrix} 3 \\ 1 \\ - 1 \\ 3\end{smallmatrix}\right], [1−133]\left[\begin{smallmatrix} 1 \\ - 1 \\ 3 \\ 3\end{smallmatrix}\right]

(5)

[1212]\left[\begin{smallmatrix} 1 \\ 2 \\ 1 \\ 2\end{smallmatrix}\right], [2301]\left[\begin{smallmatrix} 2 \\ 3 \\ 0 \\ 1\end{smallmatrix}\right], [32−12]\left[\begin{smallmatrix} 3 \\ 2 \\ -1 \\ 2\end{smallmatrix}\right], [4121]\left[\begin{smallmatrix} 4 \\ 1 \\ 2 \\ 1\end{smallmatrix}\right]

(6)

[1423]\left[\begin{smallmatrix} 1 \\ 4 \\ 2 \\ 3\end{smallmatrix}\right], [1503]\left[\begin{smallmatrix} 1 \\ 5 \\ 0 \\ 3\end{smallmatrix}\right], [−1927]\left[\begin{smallmatrix} -1 \\ 9 \\ 2 \\ 7\end{smallmatrix}\right]

(7)

[4−2−10],[9−2−20],[−3−1111]\left[\begin{smallmatrix} 4 \\ -2 \\ -1 \\ 0\end{smallmatrix}\right],\left[\begin{smallmatrix} 9 \\ -2 \\ -2 \\ 0\end{smallmatrix}\right],\left[\begin{smallmatrix} -3 \\ -1 \\ 11 \\ 1\end{smallmatrix}\right]

(8)

[1213],[4326],[43−74]\left[\begin{smallmatrix} 1 \\ 2 \\ 1 \\ 3\end{smallmatrix}\right],\left[\begin{smallmatrix} 4 \\ 3 \\ 2 \\ 6\end{smallmatrix}\right],\left[\begin{smallmatrix} 4 \\ 3 \\ -7 \\ 4\end{smallmatrix}\right]

Problem 26.44

Orthogonalize and normalize the system of vectors given in a basis ee by their coordinate columns. The Gram matrix Γ\Gamma of the basis ee is given:

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(1)

[13],[24],Γ=A19\left[\begin{smallmatrix} 1 \\ 3\end{smallmatrix}\right],\left[\begin{smallmatrix} 2 \\ 4\end{smallmatrix}\right], \quad \Gamma = \mathbf{A}_{19}. A19=[1112]\mathbf{A}_{19} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 2 \end{smallmatrix}\right].

(2)

[120]\left[\begin{smallmatrix} 1 \\ 2 \\ 0\end{smallmatrix}\right], [203]\left[\begin{smallmatrix} 2 \\ 0 \\ 3\end{smallmatrix}\right], [186]\left[\begin{smallmatrix} 1 \\ 8 \\ 6\end{smallmatrix}\right], Γ=A387\Gamma = \mathbf{A}_{387}. A387=[210121012]\mathbf{A}_{387} = \left[\begin{smallmatrix} 2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 2 \end{smallmatrix}\right].

(3)

[111]\left[\begin{smallmatrix} 1 \\ 1 \\ 1\end{smallmatrix}\right], [421]\left[\begin{smallmatrix} 4 \\ 2 \\ 1\end{smallmatrix}\right], [19−8]\left[\begin{smallmatrix} 1 \\ 9 \\ - 8\end{smallmatrix}\right], Γ=A386\Gamma = \mathbf{A}_{386}. A386=[20−1021−112]\mathbf{A}_{386} = \left[\begin{smallmatrix} 2 & 0 & -1 \\ 0 & 2 & 1 \\ -1 & 1 & 2 \end{smallmatrix}\right].

(4)

[−321]\left[\begin{smallmatrix} - 3 \\ 2 \\ 1\end{smallmatrix}\right], [−854]\left[\begin{smallmatrix} - 8 \\ 5 \\ 4\end{smallmatrix}\right], [240]\left[\begin{smallmatrix} 2 \\ 4 \\ 0\end{smallmatrix}\right], Γ=A207\Gamma = \mathbf{A}_{207}. A207=[12025−20−25]\mathbf{A}_{207} = \left[\begin{smallmatrix} 1 & 2 & 0 \\ 2 & 5 & -2 \\ 0 & -2 & 5 \end{smallmatrix}\right].

Problem 26.45
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(1)

Using the orthogonalization process, prove that a nondegenerate square matrix can be decomposed into the product of an orthogonal matrix QQ and an upper triangular matrix RR with positive diagonal elements (QR decomposition).

(2)

Prove the uniqueness of the QRQR decomposition of a nondegenerate matrix.

Problem 26.46

Find the QRQR decomposition of the given matrix (problem 26. 45):

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(1)

[1−216]\left[\begin{smallmatrix} 1 & - 2 \\ 1 & 6\end{smallmatrix}\right]

(2)

[391−7]\left[\begin{smallmatrix} 3 & 9 \\ 1 & - 7\end{smallmatrix}\right]

(3)

[12010001−1]\left[\begin{smallmatrix} 1 & 2 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & - 1\end{smallmatrix}\right]

(4)

[1341−12−22−22−2]\left[\begin{smallmatrix} 1 & 3 & 4 \\ 1 & - 1 & 2 \\ - 2\sqrt{2} & - 2\sqrt{2} & - \sqrt{2}\end{smallmatrix}\right]

(5)

[1331131−111−13−1−133]\left[\begin{smallmatrix} 1 & 3 & 3 & 1 \\ 1 & 3 & 1 & - 1 \\ 1 & 1 & - 1 & 3 \\ - 1 & - 1 & 3 & 3\end{smallmatrix}\right]

(6)

[1234232110−122121]\left[\begin{smallmatrix} 1 & 2 & 3 & 4 \\ 2 & 3 & 2 & 1 \\ 1 & 0 & - 1 & 2 \\ 2 & 1 & 2 & 1\end{smallmatrix}\right].

Problem 26.47

Prove that the Gram matrix of an arbitrary basis can be decomposed into the product Γ=RTR\Gamma = R^{T} R, where RR is an upper triangular matrix with positive elements on the diagonal. (Compare this with problem 25. 29.)

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Problem 26.48

In a Euclidean space the Gram matrix of a basis f\mathbf{f} is known. Find the transition matrix to the orthonormal basis obtained by orthogonalizing f\mathbf{f}:

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(1)

Γ=[25513]\Gamma =\left[\begin{smallmatrix} 2 & 5 \\ 5 & 13\end{smallmatrix}\right]

(2)

Γ=[13313]\Gamma =\left[\begin{smallmatrix} 1 & 3 \\ 3 & 13\end{smallmatrix}\right]

(3)

Γ=[1232583814]\Gamma =\left[\begin{smallmatrix} 1 & 2 & 3 \\ 2 & 5 & 8 \\ 3 & 8 & 14\end{smallmatrix}\right]

(4)

Γ=[40040440048040020]\Gamma =\left[\begin{smallmatrix} 4 & 0 & 0 & 4 \\ 0 & 4 & 4 & 0 \\ 0 & 4 & 8 & 0 \\ 4 & 0 & 0 & 20\end{smallmatrix}\right].

Subsection 26.4Volume (26.49-26.54)[6/67%]
Problem 26.49

Prove that the determinant of the Gram matrix of a system of vectors does not change under orthogonalization of this system (without normalizing the vectors).

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Problem 26.50

Find the volume of the parallelepiped built on the given vectors. The coordinate columns of the vectors in an orthonormal basis form the matrix:

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(1)

A617\mathbf{A}_{617} for n=4n = 4. A617=[111…11011…11………………000…11000…01]\mathbf{A}_{617} = \left[\begin{smallmatrix} 1 & 1 & 1 & \ldots & 1 & 1 \\ 0 & 1 & 1 & \ldots & 1 & 1 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & 1 & 1 \\ 0 & 0 & 0 & \ldots & 0 & 1 \end{smallmatrix}\right].

(2)

[111111−1110−11]\left[\begin{smallmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ - 1 & 1 & 1 \\ 0 & - 1 & 1\end{smallmatrix}\right]

(3)

A395\mathbf{A}_{395}. A395=[143032010021]\mathbf{A}_{395} = \left[\begin{smallmatrix} 1 & 4 & 3 \\ 0 & 3 & 2 \\ 0 & 1 & 0 \\ 0 & 2 & 1 \end{smallmatrix}\right].

(4)

A397\mathbf{A}_{397}. A397=[013235357340]\mathbf{A}_{397} = \left[\begin{smallmatrix} 0 & 1 & 3 \\ 2 & 3 & 5 \\ 3 & 5 & 7 \\ 3 & 4 & 0 \end{smallmatrix}\right].

(5)

A398\mathbf{A}_{398}. A398=[−1−1−1100010001]\mathbf{A}_{398} = \left[\begin{smallmatrix} -1 & -1 & -1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{smallmatrix}\right].

(6)

A412\mathbf{A}_{412}. A412=[00111−5100010003]\mathbf{A}_{412} = \left[\begin{smallmatrix} 0 & 0 & 1 \\ 1 & 1 & -5 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 3 \end{smallmatrix}\right].

(7)

A436\mathbf{A}_{436}. A436=[1111101100100011]\mathbf{A}_{436} = \left[\begin{smallmatrix} 1 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{smallmatrix}\right].

Problem 26.51

Find the volume of the parallelepiped built on the given vectors. The coordinate columns of the vectors in a basis e form the matrix AA. The Gram matrix of the basis ee equals Γ\Gamma:

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(1)

A=A402,Γ=[2100121001210012]A = \mathbf{A}_{402}, \quad \Gamma =\left[\begin{smallmatrix} 2 & 1 & 0 & 0 \\ 1 & 2 & 1 & 0 \\ 0 & 1 & 2 & 1 \\ 0 & 0 & 1 & 2\end{smallmatrix}\right]. A402=[100100010001]\mathbf{A}_{402} = \left[\begin{smallmatrix} 1 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{smallmatrix}\right].

(2)

A=A404,Γ=A630A = \mathbf{A}_{404}, \quad \Gamma = \mathbf{A}_{630} for n=4n = 4. A404=[100010001−2−33],A630=[111…11121…11113…11………………111…1n]\mathbf{A}_{404} = \left[\begin{smallmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ -2 & -3 & 3 \end{smallmatrix}\right], \mathbf{A}_{630} = \left[\begin{smallmatrix} 1 & 1 & 1 & \ldots & 1 & 1 \\ 1 & 2 & 1 & \ldots & 1 & 1 \\ 1 & 1 & 3 & \ldots & 1 & 1 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 1 & 1 & 1 & \ldots & 1 & n \end{smallmatrix}\right].

(3)

A=A397,Γ=A424A = \mathbf{A}_{397}, \quad \Gamma = \mathbf{A}_{424}. A397=[013235357340],A424=[200−102−100−120−1002]\mathbf{A}_{397} = \left[\begin{smallmatrix} 0 & 1 & 3 \\ 2 & 3 & 5 \\ 3 & 5 & 7 \\ 3 & 4 & 0 \end{smallmatrix}\right], \mathbf{A}_{424} = \left[\begin{smallmatrix} 2 & 0 & 0 & -1 \\ 0 & 2 & -1 & 0 \\ 0 & -1 & 2 & 0 \\ -1 & 0 & 0 & 2 \end{smallmatrix}\right].

Problem 26.52

Prove that the volume of the parallelepiped built on the vectors f1,…,fnf_{1}, \ldots , f_{n}, does not exceed the product of the lengths of its edges:

V{f1,…,fn}=det⁡Γ(f1,…,fn)⩽∣f1∣2…∣fn∣2, V\left\{ f_{1}, \ldots , f_{n}\right\} = \operatorname {det} \Gamma \left(f_{1}, \ldots , f_{n}\right) \leqslant \left|f_{1}\right|^{2} \ldots \left|f_{n}\right|^{2},

and equality holds if and only if the edges are pairwise orthogonal.

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Problem 26.53

For a square matrix AA of order nn:

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(1)

prove Hadamard's inequality

∣det⁡A∣2⩽∏k=1n(∑i=1n∣aik∣2) |\operatorname {det} A|^{2} \leqslant \prod _{k = 1}^{n}\left(\sum _{i = 1}^{n}\left|a_{ik}\right|^{2}\right)
(2)

determine the conditions under which Hadamard's inequality holds as an equality

(3)

write out Hadamard's inequality for the matrix A18\mathbf{A}_{18}. What explains such a large difference between the right and left sides? A18=[1234]\mathbf{A}_{18} = \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \end{smallmatrix}\right].

Problem 26.54
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(1)

Let e1,…,ene_{1}, \ldots , e_{n} be a basis in a Euclidean space, and let ek+1′′,…,en′′e_{k + 1}^{\prime \prime }, \ldots , e_{n}^{\prime \prime } be the orthogonal projections of the vectors ek+1,……,ene_{k + 1}, \ldots \ldots , e_{n} onto the orthogonal complement of the linear span of e1,…,eke_{1}, \ldots , e_{k}. Prove that V{e1,…,en}=V{e1,…,ek}V{ek+1′′,…,en′′}V\left\{ e_{1}, \ldots , e_{n}\right\} = V\left\{ e_{1}, \ldots , e_{k}\right\} V\left\{ e_{k + 1}^{\prime \prime }, \ldots , e_{n}^{\prime \prime }\right\}.

(2)

In an nn-dimensional Euclidean space a subspace L\mathcal{L} and linearly independent vectors a1,…,apa_{1}, \ldots , a_{p} are given. Denote by a1′,…,ap′a_{1}^{\prime }, \ldots , a_{p}^{\prime } the orthogonal projections of these vectors onto L\mathcal{L}. Prove that det⁡Γ(a1,…,ap)⩾det⁡Γ(a1′,…,ap′)\operatorname {det} \Gamma \left(a_{1}, \ldots , a_{p}\right) \geqslant \operatorname {det} \Gamma \left(a_{1}^{\prime }, \ldots , a_{p}^{\prime }\right).

(3)

Prove that the volume of the parallelepiped V{f1,…,fn}V\left\{ f_{1}, \ldots , f_{n}\right\}, built on the vectors f1,…,fnf_{1}, \ldots , f_{n}, does not exceed the product of the volumes V{f1,…,fk}V\left\{ f_{1}, \ldots , f_{k}\right\} and V{fk+1,…,fn}V\left\{ f_{k + 1}, \ldots , f_{n}\right\}.

Subsection 26.5The angle between a vector and a subspace (26.55-26.58)[4/25%]
Problem 26.55

Let x′x^{\prime } be the orthogonal projection of the vector xx onto the subspace L\mathcal{L}. Prove that the angle between the vector xx and the subspace L\mathcal{L} equals the angle between xx and x′x^{\prime } if x′≠ox^{\prime } \neq o, and equals π/2\pi / 2 if x′=ox^{\prime } = o.

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Problem 26.56

Let x′x^{\prime } be the orthogonal projection of the vector xx onto the subspace L\mathcal{L}, and let the angle between the vector xx and the subspace L\mathcal{L} equal φ\varphi. Prove that cos⁡φ=∣x′∣/∣x∣\cos \varphi = \left|x^{\prime }\right| / \left|x\right|.

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Problem 26.57

Prove that the sum of the angles of the vector xx with the subspaces L\mathcal{L} and L⊥\mathcal{L}^{\perp } equals π/2\pi / 2.

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Problem 26.58

In an orthonormal basis the vectors xx and f1,…,fkf_{1}, \ldots , f_{k} are given by their coordinate columns ξ\xi and φ1,…,φk\varphi_{1}, \ldots , \varphi_{k}. Find the angle between the vector xx and the subspace L\mathcal{L} spanned by xx and f1,…,fkf_{1}, \ldots , f_{k}:

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(1)

ξ=[1212],φ1=[50−42],φ2=[31−50]\xi =\left[\begin{smallmatrix} 1 \\ 2 \\ 1 \\ 2\end{smallmatrix}\right], \varphi_{1} =\left[\begin{smallmatrix} 5 \\ 0 \\ - 4 \\ 2\end{smallmatrix}\right], \varphi_{2} =\left[\begin{smallmatrix} 3 \\ 1 \\ - 5 \\ 0\end{smallmatrix}\right];

(2)

ξ=[1133],φ1=[−4423],φ2=[−5210]\xi =\left[\begin{smallmatrix} 1 \\ 1 \\ 3 \\ 3\end{smallmatrix}\right], \varphi_{1} =\left[\begin{smallmatrix} -4 \\ 4 \\ 2 \\ 3\end{smallmatrix}\right], \varphi_{2} =\left[\begin{smallmatrix} -5 \\ 2 \\ 1 \\ 0\end{smallmatrix}\right];

(3)

ξ=[2626],φ1=[0−1−11],φ2=[101−1],φ3=[−3110]\xi =\left[\begin{smallmatrix} 2 \\ 6 \\ 2 \\ 6\end{smallmatrix}\right], \varphi_{1} =\left[\begin{smallmatrix} 0 \\ - 1 \\ - 1 \\ 1\end{smallmatrix}\right], \varphi_{2} =\left[\begin{smallmatrix} 1 \\ 0 \\ 1 \\ - 1\end{smallmatrix}\right], \varphi_{3} =\left[\begin{smallmatrix} -3 \\ 1 \\ 1 \\ 0\end{smallmatrix}\right];

Subsection 26.6Reflection (26.59-26.62)[4/100%]
Problem 26.59

Let nonzero vectors xx and aa be given by their coordinate columns ξ\mathbf{\xi } and a\mathbf{a} respectively, and let the subspace L\mathcal{L} be defined by the equation (a,x)=0(a, x) = 0. Find the image yy under reflection of the vector xx in the subspace L\mathcal{L}:

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(1)

ξ=[11],a=[10]\xi = \left[\begin{smallmatrix} 1 \\ 1 \end{smallmatrix}\right], \quad \mathbf{a} = \left[\begin{smallmatrix} 1 \\ 0 \end{smallmatrix}\right], the basis is orthonormal

(2)

ξ=c162,a=c171\xi = \mathbf{c}_{162}, \quad \mathbf{a} = \mathbf{c}_{171}, the basis is orthonormal. c162=[9036],c171=[−1110]\mathbf{c}_{162} = \left[\begin{smallmatrix} 9 \\ 0 \\ 3 \\ 6 \end{smallmatrix}\right], \mathbf{c}_{171} = \left[\begin{smallmatrix} -1 \\ 1 \\ 1 \\ 0 \end{smallmatrix}\right].

(3)

ξ=[11],a=[10]\xi = \left[\begin{smallmatrix} 1 \\ 1 \end{smallmatrix}\right], \quad \mathbf{a} = \left[\begin{smallmatrix} 1 \\ 0 \end{smallmatrix}\right], the basis has Gram matrix A19\mathbf{A}_{19}. A19=[1112]\mathbf{A}_{19} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 2 \end{smallmatrix}\right].

(4)

ξ=[221−1],a=[1100]\xi =\left[\begin{smallmatrix} 2 \\ 2 \\ 1 \\ - 1\end{smallmatrix}\right], \quad \mathbf{a} =\left[\begin{smallmatrix} 1 \\ 1 \\ 0 \\ 0\end{smallmatrix}\right], the basis has Gram matrix [4004044004804008]\left[\begin{smallmatrix} 4 & 0 & 0 & 4 \\ 0 & 4 & 4 & 0 \\ 0 & 4 & 8 & 0 \\ 4 & 0 & 0 & 8\end{smallmatrix}\right].

Problem 26.60

In an orthonormal basis, a subspace L\mathcal{L} is given by a system of linear equations with matrix AA, and a vector xx -- by its coordinate column ξ\xi. Find the image yy under reflection of the vector xx in the subspace L\mathcal{L}:

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(1)

A=[111],ξ=[123]A =\left[\begin{smallmatrix} 1 & 1 & 1\end{smallmatrix}\right], \quad \xi =\left[\begin{smallmatrix} 1 \\ 2 \\ 3\end{smallmatrix}\right]

(2)

A=[32−5−1−12],ξ=[426]A =\left[\begin{smallmatrix} 3 & 2 & - 5 \\ - 1 & - 1 & 2\end{smallmatrix}\right], \quad \xi =\left[\begin{smallmatrix} 4 \\ 2 \\ 6\end{smallmatrix}\right]

(3)

A=[11111],ξ=[54321]A =\left[\begin{smallmatrix} 1 & 1 & 1 & 1 & 1\end{smallmatrix}\right], \quad \xi =\left[\begin{smallmatrix} 5 \\ 4 \\ 3 \\ 2 \\ 1\end{smallmatrix}\right]

(4)

A=[2−3513−561],ξ=[7−594]A =\left[\begin{smallmatrix} 2 & - 3 & 5 & 1 \\ 3 & - 5 & 6 & 1\end{smallmatrix}\right], \quad \xi =\left[\begin{smallmatrix} 7 \\ - 5 \\ 9 \\ 4\end{smallmatrix}\right]

(5)

A=[435−2324−1],ξ=[−2420]A =\left[\begin{smallmatrix} 4 & 3 & 5 & -2 \\ 3 & 2 & 4 & -1\end{smallmatrix}\right], \quad \xi =\left[\begin{smallmatrix} -2 \\ 4 \\ 2 \\ 0\end{smallmatrix}\right]

(6)

A=[1031341018311],ξ=[−2420]A =\left[\begin{smallmatrix} 10 & 3 & 1 & 3 \\ 4 & 1 & 0 & 1 \\ 8 & 3 & 1 & 1\end{smallmatrix}\right], \quad \xi = \left[\begin{smallmatrix} -2 \\ 4 \\ 2 \\ 0\end{smallmatrix}\right].

Problem 26.61

Under what necessary and sufficient condition can the vector xx be carried into the vector yy by means of a reflection in an (n−1)(n - 1)-dimensional subspace L\mathcal{L}? How can such a subspace be found if the condition holds?

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Problem 26.62

Find an (n−1)(n - 1)-dimensional subspace L\mathcal{L} such that the vector xx, under reflection in it, is carried into the given vector yy. The vectors are given in an orthonormal basis by their coordinate columns ξ\xi and η\eta:

?
(1)

ξ=[12],η=[21]\xi =\left[\begin{smallmatrix} 1 \\ 2\end{smallmatrix}\right], \quad \eta = \left[\begin{smallmatrix} 2 \\ 1 \end{smallmatrix}\right]

(2)

ξ=c192,η=c197\xi = \mathbf{c}_{192}, \quad \eta = \mathbf{c}_{197}. c192=[1030],c197=[1122]\mathbf{c}_{192} = \left[\begin{smallmatrix} 1 \\ 0 \\ 3 \\ 0 \end{smallmatrix}\right], \mathbf{c}_{197} = \left[\begin{smallmatrix} 1 \\ 1 \\ 2 \\ 2 \end{smallmatrix}\right].

(3)

ξ=c160,η=c182\xi = \mathbf{c}_{160}, \quad \eta = \mathbf{c}_{182}. c160=[4530],c182=[1001]\mathbf{c}_{160} = \left[\begin{smallmatrix} 4 \\ 5 \\ 3 \\ 0 \end{smallmatrix}\right], \mathbf{c}_{182} = \left[\begin{smallmatrix} 1 \\ 0 \\ 0 \\ 1 \end{smallmatrix}\right].

(4)

ξ=c203,η=c168\xi = \mathbf{c}_{203}, \quad \eta = \mathbf{c}_{168}. c203=[2222],c168=[0100]\mathbf{c}_{203} = \left[\begin{smallmatrix} 2 \\ 2 \\ 2 \\ 2 \end{smallmatrix}\right], \mathbf{c}_{168} = \left[\begin{smallmatrix} 0 \\ 1 \\ 0 \\ 0 \end{smallmatrix}\right].

Subsection 26.7Linear functions on Euclidean space (26.63-26.74)[12/83%]
Problem 26.63

Find the coefficients of the linear function associated with the given vector. The vector is given by the coordinate column a in a basis with Gram matrix Γ\Gamma:

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(1)

a=[111],Γ=A294\mathbf{a} =\left[\begin{smallmatrix} 1 \\ 1 \\ 1\end{smallmatrix}\right], \Gamma = \mathbf{A}_{294}. A294=[2−20−292022]\mathbf{A}_{294} = \left[\begin{smallmatrix} 2 & -2 & 0 \\ -2 & 9 & 2 \\ 0 & 2 & 2 \end{smallmatrix}\right].

(2)

a=[11−1],Γ=A308\mathbf{a} =\left[\begin{smallmatrix} 1 \\ 1 \\ - 1\end{smallmatrix}\right], \Gamma = \mathbf{A}_{308}. A308=[5−12−110204]\mathbf{A}_{308} = \left[\begin{smallmatrix} 5 & -1 & 2 \\ -1 & 1 & 0 \\ 2 & 0 & 4 \end{smallmatrix}\right].

(3)

a=[−210],Γ=A207\mathbf{a} =\left[\begin{smallmatrix} - 2 \\ 1 \\ 0\end{smallmatrix}\right], \quad \Gamma = \mathbf{A}_{207}. A207=[12025−20−25]\mathbf{A}_{207} = \left[\begin{smallmatrix} 1 & 2 & 0 \\ 2 & 5 & -2 \\ 0 & -2 & 5 \end{smallmatrix}\right].

Problem 26.64

Find the coordinate column of the vector associated with the given linear function. The function is given by the row of coefficients φ\varphi in a basis with Gram matrix Γ\Gamma:

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(1)

φ=[404],Γ=A294\varphi =\left[\begin{smallmatrix} 4 & 0 & 4\end{smallmatrix}\right], \quad \Gamma = \mathbf{A}_{294}. A294=[2−20−292022]\mathbf{A}_{294} = \left[\begin{smallmatrix} 2 & -2 & 0 \\ -2 & 9 & 2 \\ 0 & 2 & 2 \end{smallmatrix}\right].

(2)

φ=[402],Γ=A308\varphi =\left[\begin{smallmatrix} 4 & 0 & 2\end{smallmatrix}\right], \quad \Gamma = \mathbf{A}_{308}. A308=[5−12−110204]\mathbf{A}_{308} = \left[\begin{smallmatrix} 5 & -1 & 2 \\ -1 & 1 & 0 \\ 2 & 0 & 4 \end{smallmatrix}\right].

(3)

φ=[353],Γ=A207\varphi =\left[\begin{smallmatrix} 3 & 5 & 3\end{smallmatrix}\right], \quad \Gamma = \mathbf{A}_{207}. A207=[12025−20−25]\mathbf{A}_{207} = \left[\begin{smallmatrix} 1 & 2 & 0 \\ 2 & 5 & -2 \\ 0 & -2 & 5 \end{smallmatrix}\right].

Problem 26.65

In the space of square matrices of order nn with the scalar product (X,Y)=tr⁡XTY(X, Y) = \operatorname {tr} X^{T} Y, find the vector (matrix) CC associated with the function:

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(1)

f(X)=tr⁡Xf(X) = \operatorname {tr} X;

(2)

f(X)=∑i,jxijf(X) = \sum_{i, j} x_{ij};

(3)

n=4n = 4 and f(X)f(X) equals the entry of the product of the matrix A444\mathbf{A}_{444} with XX located in the first row and first column. A444=[54132114321113−22]\mathbf{A}_{444} = \left[\begin{smallmatrix} 5 & 4 & 1 & 3 \\ 2 & 1 & 1 & 4 \\ 3 & 2 & 1 & 1 \\ 1 & 3 & -2 & 2 \end{smallmatrix}\right].

Problem 26.66

In the space of polynomials of degree ⩽3\leqslant 3 with the standard scalar product, a linear function assigns to a polynomial p(t)p(t) its constant term p(0)p(0). Find the vector (polynomial) associated with this linear function.

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Problem 26.67

The vectors in a basis ee are permuted. How does its biorthogonal basis change?

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Problem 26.68

In an arithmetic space with the standard scalar product, the columns ξ1,…,ξn\xi_{1}, \ldots , \xi_{n} form a basis. Find the corresponding biorthogonal basis:

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(1)

ξ1=[20],ξ2=[03]\xi_{1} = \left[\begin{smallmatrix} 2 \\ 0 \end{smallmatrix}\right], \quad \xi_{2} =\left[\begin{smallmatrix} 0 \\ 3\end{smallmatrix}\right]

(2)

ξ1=[12],ξ2=[21]\xi_{1} = \left[\begin{smallmatrix} 1 \\ 2 \end{smallmatrix}\right], \quad \xi_{2} = \left[\begin{smallmatrix} 2 \\ 1 \end{smallmatrix}\right]

(3)

ξ1=[13],ξ2=[25]\xi_{1} = \left[\begin{smallmatrix} 1 \\ 3 \end{smallmatrix}\right], \quad \xi_{2} = \left[\begin{smallmatrix} 2 \\ 5 \end{smallmatrix}\right]

(4)

ξ1=[111],ξ2=[110],ξ3=[100]\xi_{1} =\left[\begin{smallmatrix} 1 \\ 1 \\ 1\end{smallmatrix}\right], \quad \xi_{2} =\left[\begin{smallmatrix} 1 \\ 1 \\ 0\end{smallmatrix}\right], \quad \xi_{3} =\left[\begin{smallmatrix} 1 \\ 0 \\ 0\end{smallmatrix}\right].

Problem 26.69

In a Euclidean space, in a basis ee with Gram matrix Γ\Gamma, the coordinates of the vectors of a basis h\mathbf{h} are given. Find the coordinates of the vectors of the biorthogonal basis h∗\mathbf{h}^{ * }:

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(1)

[13],[25],Γ=A19\left[\begin{smallmatrix} 1 \\ 3 \end{smallmatrix}\right], \quad \left[\begin{smallmatrix} 2 \\ 5 \end{smallmatrix}\right], \quad \Gamma = \mathbf{A}_{19}. A19=[1112]\mathbf{A}_{19} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 2 \end{smallmatrix}\right].

(2)

[1−1],[11],Γ=A32\left[\begin{smallmatrix} 1 \\ -1 \end{smallmatrix}\right], \quad \left[\begin{smallmatrix} 1 \\ 1 \end{smallmatrix}\right], \quad \Gamma = \mathbf{A}_{32}. A32=[13310]\mathbf{A}_{32} = \left[\begin{smallmatrix} 1 & 3 \\ 3 & 10 \end{smallmatrix}\right].

(3)

[100],[110],[111],Γ=A387\left[\begin{smallmatrix} 1 \\ 0 \\ 0\end{smallmatrix}\right], \quad \left[\begin{smallmatrix} 1 \\ 1 \\ 0\end{smallmatrix}\right], \quad \left[\begin{smallmatrix} 1 \\ 1 \\ 1\end{smallmatrix}\right], \quad \Gamma = \mathbf{A}_{387}. A387=[210121012]\mathbf{A}_{387} = \left[\begin{smallmatrix} 2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 2 \end{smallmatrix}\right].

(4)

[011],[121],[−111],Γ=A385\left[\begin{smallmatrix} 0 \\ 1 \\ 1\end{smallmatrix}\right], \quad \left[\begin{smallmatrix} 1 \\ 2 \\ 1\end{smallmatrix}\right], \quad \left[\begin{smallmatrix} -1 \\ 1 \\ 1\end{smallmatrix}\right], \quad \Gamma = \mathbf{A}_{385}.

Problem 26.70

In the space of polynomials of degree at most 2 with the standard scalar product, find the basis biorthogonal to the basis 1,t,t21, t, t^{2}.

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Problem 26.71

Prove that the coordinates of the vector xx in the basis ee can be computed by the formulas ξi=(x,ei∗),i=1,…,n\xi^{i} = \left(x, e_{i}^{ * }\right), \quad i = 1, \ldots , n, where ei∗e_{i}^{ * } are the vectors of the basis biorthogonal to e.

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Problem 26.72

Prove that the scalar product in a Euclidean space can be computed by the formula (x,y)=ξ1η1∗+……+ξnηn∗(x, y) = \xi^{1} \eta_{1}^{ * } + \ldots \ldots + \xi^{n} \eta_{n}^{ * }, where ξ1,…,ξn\xi_{1}, \ldots , \xi_{n} are the coordinates of the vector xx in the basis e\mathbf{e}, and η1∗,…,ηn∗\eta_{1}^{ * }, \ldots , \eta_{n}^{ * } are the coordinates of the vector yy in the biorthogonal basis e∗\mathbf{e}^{ * }.

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Problem 26.73
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(1)

Find the transition matrix from the basis ee to its biorthogonal basis e∗\mathbf{e}^{ * }.

(2)

Using the result obtained, prove that Γe∗=Γe−1\Gamma_{\mathbf{e}^{ * }} = \Gamma_{\mathbf{e}}^{ - 1}.

Problem 26.74

Let SS be the transition matrix from the basis e\mathbf{e} to the basis f\mathbf{f}. Find the transition matrix from the basis e∗\mathbf{e}^{ * }, biorthogonal to e\mathbf{e}, to the basis f∗\mathbf{f}^{ * }, biorthogonal to f\mathbf{f}.

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