Subsection 26.7

Linear functions on Euclidean space (26.63-26.74)

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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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