Math Problems Vault

Problem 28.1

(1)(2)

Let $a_{1}, \ldots , a_{k}$ be an orthonormal basis in the subspace $\mathcal{L} \subset \mathcal{E}$ . The coordinate columns of these vectors in an orthonormal basis $e$ of the space $\mathcal{E}$ form the matrix $A$ . Write, in the basis $e$ , the matrix of

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(1)

orthogonal projection onto the space $\mathcal{L}$

(2)

the matrix of reflection in the subspace $\mathcal{L}$ .

Problem 28.2

(1)(2)

Let $a_{1}, \ldots , a_{k}$ be a basis in the subspace $\mathcal{L} \subset \mathcal{E}$ . The coordinate columns of these vectors in an orthonormal basis $e$ of the space $\mathcal{E}$ form the matrix $A$ . Write, in the basis $e$ , the matrix:

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(1)

of orthogonal projection onto the subspace $\mathcal{L}$

(2)

the matrix of reflection in the subspace $\mathcal{L}$ .

Problem 28.3

(1)(2)

Let $a_{1}, \ldots , a_{k}$ be a basis in the subspace $\mathcal{L} \subset \mathcal{E}$ . The coordinate columns of these vectors in a basis $e$ of the space $\mathcal{E}$ form the matrix $A$ . The Gram matrix $\Gamma$ of the basis $\mathbf{e}$ is given. Write, in the basis $e$ , the matrix:

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(1)

of orthogonal projection onto the subspace $\mathcal{L}$

(2)

the matrix of reflection in the subspace $\mathcal{L}$ .

Problem 28.4

(1)(2)

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(1)

A vector $a$ is given, and the subspace $\mathcal{L} \subset \mathcal{E}$ is defined by the equation $(a, x) = 0$ . Find the image of the vector $x$ under reflection in $\mathcal{L}$ and express the matrix of this transformation in terms of the coordinate column a of the vector $a$ in an orthonormal basis.

(2)

Let a vector $x$ and a vector $y$ of length 1 be given. Find an $(n - 1)$ -dimensional subspace, under reflection in which $x$ is carried into the vector $\lambda y\quad (\lambda > 0)$ . What is $\lambda$ equal to?

Problem 28.5

Find the matrix of reflection in an $(n - 1)$ -dimensional subspace that carries the vector $x$ into the vector $y$ . The vectors are given by their coordinate columns $\xi$ and $\eta$ in an orthonormal basis:

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(1)

$\xi =\left[\begin{smallmatrix} 1 \\ 2\end{smallmatrix}\right], \; \eta =\left[\begin{smallmatrix} 2 \\ 1\end{smallmatrix}\right]$

(2)

$\xi =\left[\begin{smallmatrix} -1 \\ 2\end{smallmatrix}\right], \; \eta =\left[\begin{smallmatrix} 1 \\ -2\end{smallmatrix}\right]$

(3)

$\xi =\left[\begin{smallmatrix} 1 \\ 1 \\ 1\end{smallmatrix}\right], \; \eta =\left[\begin{smallmatrix} 1 \\ - 1 \\ 1\end{smallmatrix}\right]$

(4)

$\xi =\left[\begin{smallmatrix} 2 \\ 1 \\ - 2\end{smallmatrix}\right], \; \eta =\left[\begin{smallmatrix} 0 \\ 3 \\ 0\end{smallmatrix}\right]$

(5)

$\xi =\left[\begin{smallmatrix} 1 \\ 0 \\ 2 \\ 1\end{smallmatrix}\right], \; \eta =\left[\begin{smallmatrix} 2 \\ - 1 \\ 1 \\ 0\end{smallmatrix}\right]$ .

Problem 28.6

Determine the geometric meaning of the transformation given in an orthonormal basis by the matrix:

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(1)

$(1 / 2) \mathbf{A}_{468}$ . $\mathbf{A}_{468} = \left[\begin{smallmatrix} 1 & 1 & 1 & -1 \\ 1 & 1 & -1 & 1 \\ 1 & -1 & 1 & 1 \\ -1 & 1 & 1 & 1 \end{smallmatrix}\right]$ .

(2)

$(1 / 3) \mathbf{A}_{202}$ . $\mathbf{A}_{202} = \left[\begin{smallmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ 2 & -2 & 1 \end{smallmatrix}\right]$ .

(3)

$(1 / 18) \mathbf{A}_{350}$ . $\mathbf{A}_{350} = \left[\begin{smallmatrix} 1 & -4 & 1 \\ -4 & 16 & -4 \\ 1 & -4 & 1 \end{smallmatrix}\right]$ .

(4)

$\mathbf{A}_{430}$ . $\mathbf{A}_{430} = \left[\begin{smallmatrix} 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{smallmatrix}\right]$ .

(5)

$\mathbf{A}_{431}$ . $\mathbf{A}_{431} = \left[\begin{smallmatrix} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{smallmatrix}\right]$ .

Problem 28.7

(1)(2)(3)(4)

Consider a four-dimensional Euclidean space and an orthonormal basis $e$ in it. In all cases the rotation is carried out in the direction from the first of the indicated vectors to the second. Write the matrix:

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(1)

of a rotation by $\pi / 2$ in the linear span $\mathcal{L}$ of the vectors $e_{2} + e_{3}$ and $e_{1} - e_{4}$ (the vectors of $\mathcal{L}^{\perp }$ are fixed);

(2)

of a rotation by $\pi / 4$ in the linear span $\mathcal{L}$ of the vectors $e_{1} + e_{4}$ and $e_{2} - e_{3}$ (the vectors of $\mathcal{L}^{\perp }$ are fixed);

(3)

of a rotation by $\pi / 6$ in the linear span $\mathcal{L}$ of the vectors $e_{1} - e_{2}$ and $e_{3} - e_{4}$ (the vectors of $\mathcal{L}^{\perp }$ are fixed);

(4)

of a rotation by $\pi / 4$ in the linear span $\mathcal{L}$ of the vectors $e_{1} + e_{2}$ and $e_{3} + e_{4}$ , and by $\pi / 4$ in the linear span $\mathcal{L}$ of the vectors $e_{3} - e_{4}$ and $e_{1} - e_{2}$ .

Problem 28.8

Let $f_{1}, \ldots , f_{m}$ and $g_{1}, \ldots , g_{m}$ be two systems of vectors in an $n$ -dimensional Euclidean space. Their coordinate columns in an orthonormal basis form, respectively, the matrices $F$ and $G$ . Find the matrix of the transformation

\varphi (x) = \sum _{j = 1}^{m}\left(x, f_{j}\right) g_{j}.

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Problem 28.9

In the subspaces $\mathcal{A}, \mathcal{B} \subset \mathcal{E}$ , orthonormal bases $a_{1}, \ldots , a_{k}$ and $b_{1}, \ldots , b_{m}$ are chosen. A vector $x\in \mathcal{A}$ is projected onto $\mathcal{B}$ , and then the resulting projection is projected onto $\mathcal{A}$ . This defines a transformation of the subspace $\mathcal{A}$ . Find its matrix in the basis $a_{1}, \ldots , a_{k}$ . Prove that the eigenvalues of this transformation belong to the interval $\left[0, 1\right]$ .

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