Math Problems Vault

Problem 26.22

(1)(2)

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

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(1)

onto the zero subspace;

(2)

onto $\mathcal{E}$ .

Problem 26.23

In the subspace $\mathcal{L} \subset \mathcal{E}$ a basis $a_{1}, \ldots , a_{k}$ is given. In an orthonormal basis of the space $\mathcal{E}$ the coordinate columns of these vectors form the matrix $A$ . A vector $x\in \mathcal{E}$ is given by its coordinate column $\xi$ . Find the orthogonal projections $x^{\prime }$ and $x^{\prime \prime }$ of the vector $x$ onto $\mathcal{L}$ and $\mathcal{L}^{\perp }$ .

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Problem 26.24

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

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Problem 26.25

In the space $\mathcal{E}$ a basis $e$ with Gram matrix $\Gamma$ is chosen. The subspace $\mathcal{L}$ is spanned by linearly independent vectors $a_{1}, \ldots , a_{k}$ , whose coordinate columns form the matrix $A$ . A vector $x\in \mathcal{E}$ is given by its coordinate column $\xi$ . Find the orthogonal projections $x^{\prime }$ and $x^{\prime \prime }$ of the vector $x$ onto $\mathcal{L}$ and $\mathcal{L}^{\perp }$ .

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Problem 26.26

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

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Problem 26.27

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

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(1)

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

(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)

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

(4)

$\mathbf{a}_{1} =\left[\begin{smallmatrix} 3 \\ - 2 \\ 1 \\ 1\end{smallmatrix}\right], \mathbf{a}_{2} =\left[\begin{smallmatrix} 1 \\ 0 \\ - 1 \\ 1\end{smallmatrix}\right]$ , $\xi =\left[\begin{smallmatrix} 2 \\ - 1 \\ 3 \\ - 2\end{smallmatrix}\right]$

(5)

$\mathbf{a}_{1} =\left[\begin{smallmatrix} 2 \\ 3 \\ 0 \\ 1\end{smallmatrix}\right], \mathbf{a}_{2} =\left[\begin{smallmatrix} 0 \\ 5 \\ - 2 \\ - 1\end{smallmatrix}\right], \xi =\left[\begin{smallmatrix} 6 \\ 0 \\ 4 \\ 2\end{smallmatrix}\right]$

(6)

$\mathbf{a}_{1} =\left[\begin{smallmatrix} 4 \\ 3 \\ - 3 \\ 2\end{smallmatrix}\right], \quad \mathbf{a}_{2} =\left[\begin{smallmatrix} 2 \\ 9 \\ 1 \\ - 4\end{smallmatrix}\right]$ , $\mathbf{a}_{3} =\left[\begin{smallmatrix} - 1 \\ 3 \\ 2 \\ - 3\end{smallmatrix}\right], \quad \xi =\left[\begin{smallmatrix} 2 \\ 4 \\ - 1 \\ 3\end{smallmatrix}\right]$

(7)

$\mathbf{a}_{1} =\left[\begin{smallmatrix} 3 \\ 1 \\ - 2 \\ - 2\end{smallmatrix}\right], \quad \mathbf{a}_{2} =\left[\begin{smallmatrix} 3 \\ 1 \\ - 1 \\ - 3\end{smallmatrix}\right]$ , $\mathbf{a}_{3} =\left[\begin{smallmatrix} 3 \\ - 1 \\ 0 \\ - 2\end{smallmatrix}\right], \quad \xi =\left[\begin{smallmatrix} 6 \\ 4 \\ - 2 \\ - 4\end{smallmatrix}\right]$

(8)

$\mathbf{a}_{1} =\left[\begin{smallmatrix} 3 \\ 1 \\ 1 \\ 2\end{smallmatrix}\right], \quad \mathbf{a}_{2} =\left[\begin{smallmatrix} 4 \\ 0 \\ 2 \\ 1\end{smallmatrix}\right], \mathbf{a}_{3} =\left[\begin{smallmatrix} 8 \\ 0 \\ 2 \\ 3\end{smallmatrix}\right]$ , $\xi =\left[\begin{smallmatrix} - 7 \\ 1 \\ 3 \\ 1\end{smallmatrix}\right]$

(9)

$\mathbf{a}_{1} = \mathbf{c}_{201}, \quad \mathbf{a}_{2} = \mathbf{c}_{202}, \quad \xi = \mathbf{c}_{168}$ . $\mathbf{c}_{201} = \left[\begin{smallmatrix} 1 \\ -7 \\ 5 \\ -2 \end{smallmatrix}\right], \mathbf{c}_{202} = \left[\begin{smallmatrix} 1 \\ 0 \\ 1 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{168} = \left[\begin{smallmatrix} 0 \\ 1 \\ 0 \\ 0 \end{smallmatrix}\right]$ .

(10)

$\mathbf{a}_{1} = \mathbf{c}_{176}, \quad \mathbf{a}_{2} = \mathbf{c}_{161}, \quad \xi = \mathbf{c}_{193}$ . $\mathbf{c}_{176} = \left[\begin{smallmatrix} -1 \\ 1 \\ -2 \\ -1 \end{smallmatrix}\right], \mathbf{c}_{161} = \left[\begin{smallmatrix} 1 \\ 3 \\ 0 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{193} = \left[\begin{smallmatrix} 0 \\ 0 \\ 0 \\ 1 \end{smallmatrix}\right]$ .

(11)

$\mathbf{a}_{1} = \mathbf{c}_{236}, \mathbf{a}_{2} = \mathbf{c}_{271}, \quad \mathbf{a}_{3} = \mathbf{c}_{283}, \quad \xi = \mathbf{c}_{232}$ . $\mathbf{c}_{236} = \left[\begin{smallmatrix} -2 \\ -1 \\ 0 \\ 1 \\ 2 \end{smallmatrix}\right], \mathbf{c}_{271} = \left[\begin{smallmatrix} 1 \\ -1 \\ 2 \\ -1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{283} = \left[\begin{smallmatrix} 1 \\ -2 \\ -3 \\ -2 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{232} = \left[\begin{smallmatrix} 2 \\ 2 \\ 0 \\ -2 \\ -2 \end{smallmatrix}\right]$ .

Problem 26.28

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

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(1)

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

(2)

$A =\left[\begin{smallmatrix} 3 & 2 & - 5 \\ - 1 & - 1 & 2\end{smallmatrix}\right], \quad \xi =\left[\begin{smallmatrix} 4 \\ 2 \\ 6\end{smallmatrix}\right]$

(3)

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

(4)

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

(5)

$A =\left[\begin{smallmatrix} 4 & 3 & 5 & -2 \\ 3 & 2 & 4 & - 1\end{smallmatrix}\right], \quad \xi =\left[\begin{smallmatrix} -2 \\ 4 \\ 2 \\ 0\end{smallmatrix}\right]$

(6)

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

Problem 26.29

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

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(1)

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

(2)

$A =\left[\begin{smallmatrix} 3 & 2 & - 5 \\ - 1 & - 1 & 2\end{smallmatrix}\right], \quad \xi =\left[\begin{smallmatrix} 4 \\ 2 \\ 6\end{smallmatrix}\right]$

(3)

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

(4)

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

(5)

$A =\left[\begin{smallmatrix} 4 & 3 & 5 & -2 \\ 3 & 2 & 4 & - 1\end{smallmatrix}\right], \quad \xi =\left[\begin{smallmatrix} -2 \\ 4 \\ 2 \\ 0\end{smallmatrix}\right]$

(6)

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

find the coordinates of the vector $y$ obtained by reflecting the vector $x$ in the subspace $\mathcal{L}$ .

Problem 26.30

(1)(2)

Find the orthogonal projection of the polynomial $35t^{4} + 15t^{3} - 15t^{2} - 8t + 4$ 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

(1)(2)

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

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(1)

for $n = 2$

(2)

for $n = 3$ .

Problem 26.32

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

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Problem 26.33

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

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Problem 26.34

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

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Problem 26.35

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

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Problem 26.36

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

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Problem 26.37

Let $\mathcal{L}_{i}, \quad (i = 1, \ldots , s)$ 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 $x$ onto these subspaces does not exceed $\left|x\right|^{2}$ , and that this bound is attained if $x$ belongs to the sum of the subspaces.

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Problem 26.38

(1)(2)

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

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(1)

$x_{2} = x_{1} + x^{\prime }$

(2)

$\left|x_{1}\right| \leqslant \left|x_{2}\right|$ , and for any $x$ equality holds if and only if $\mathcal{L}_{1} = \mathcal{L}_{2}$ .

Problem 26.39

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

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Problem 26.40

Let $e_{1}, \ldots , e_{n}$ be an orthonormal basis in a Euclidean space and let a system of vectors $g_{1}, \ldots , g_{n}$ be such that $\sum_{i = 1}^{n}\left|e_{i} - g_{i}\right| < 1$ . Prove that this system of vectors is linearly independent.

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Problem 26.41

Let $e_{1}, \ldots , e_{n}$ be an orthonormal basis in a Euclidean space and let a system of vectors $g_{1}, \ldots , g_{n}$ be such that $\sum_{i = 1}^{n} \cos \left(\widehat{e}_{i}, g_{i}\right) \geqslant (2n - 1) / 2$ . Prove that this system of vectors is linearly independent.

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