Math Problems Vault

Problem 29.38

Let the transformation $\varphi$ be isometric, i.e., $(\varphi (x), \varphi (y)) = (x, y)$ for any vectors $x$ and $y$ . Prove that $\varphi$ is linear and one-to-one.

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Problem 29.39

Prove that the orthogonal transformations of a Euclidean space $\mathcal{E}$ form a group with respect to the usual operation of composition of transformations.

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Problem 29.40

(1)(2)

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(1)

Verify that the sum of orthogonal transformations is not, in general, an orthogonal transformation.

(2)

Is the product of an orthogonal transformation by a number an orthogonal transformation?

Problem 29.41

Prove that for any two vectors of equal length there exists an orthogonal transformation carrying the first vector into the second.

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Problem 29.42

Prove that for any two orthonormal bases there exists an orthogonal transformation carrying the first basis into the second.

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Problem 29.43

Prove that the transformation of problem 28. 23 is orthogonal if and only if the matrix $A$ is orthogonal.

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Problem 29.44

In a Euclidean space a basis with Gram matrix $\Gamma$ is chosen. Find the condition on the matrix of a linear transformation, necessary and sufficient for this transformation to be orthogonal.

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Problem 29.45

(1)(2)

Can an orthogonal transformation, in some basis, have the matrix:

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(1)

$\mathbf{A}_{16}$ . $\mathbf{A}_{16} = \left[\begin{smallmatrix} 1 & -1 \\ 1 & 1 \end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{34}$ ? $\mathbf{A}_{34} = \left[\begin{smallmatrix} 2 & 1 \\ 1 & 1 \end{smallmatrix}\right]$ .

Problem 29.46

An orthogonal transformation of an arithmetic space with the standard scalar product carries the columns of the matrix $A$ into the columns of $B$ . How are the matrices $A$ and $B$ related?

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Problem 29.47

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

A linear transformation $\varphi$ of an arithmetic space with the standard scalar product carries the columns of the matrix $A$ into the columns of the matrix $B$ . Is $\varphi$ orthogonal:

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(1)

$A =\left[\begin{smallmatrix} 4 & 2 \\ 7 & 1\end{smallmatrix}\right], \quad B =\left[\begin{smallmatrix} 8 & 2 \\ 1 & - 1\end{smallmatrix}\right]$ ;

(2)

$A = \mathbf{A}_{44}, \quad B = \mathbf{A}_{34}$ . $\mathbf{A}_{44} = \left[\begin{smallmatrix} 5 & -1 \\ 4 & 1 \end{smallmatrix}\right], \mathbf{A}_{34} = \left[\begin{smallmatrix} 2 & 1 \\ 1 & 1 \end{smallmatrix}\right]$ .

(3)

$A =\left[\begin{smallmatrix} 1 & 1 & 3 \\ 0 & - 1 & - 6 \\ - 1 & 1 & 0\end{smallmatrix}\right], \quad B =\left[\begin{smallmatrix} 0 & 1 & 0 \\ 1 & - 1 & - 3 \\ - 1 & - 1 & - 6\end{smallmatrix}\right]$

(4)

$A = \mathbf{A}_{332}, \quad B = \mathbf{A}_{339}$ ? $\mathbf{A}_{332} = \left[\begin{smallmatrix} 1 & 1 & 0 \\ -1 & 1 & 1 \\ 2 & 0 & 1 \end{smallmatrix}\right], \mathbf{A}_{339} = \left[\begin{smallmatrix} 0 & -1 & 1 \\ -1 & 0 & 2 \\ 1 & -1 & 1 \end{smallmatrix}\right]$ .

Problem 29.48

Let $x_{1}, \ldots , x_{k}$ and $y_{1}, \ldots , y_{k}$ be vectors of an $n$ -dimensional Euclidean space, and let an orthogonal transformation $\varphi$ be such that $\varphi \left(x_{i}\right) = y_{i}, \quad i = 1, \ldots , k$ . Prove that this is possible if and only if the matrices of pairwise scalar products of the two systems of vectors are equal.

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Problem 29.49

Let $\mathcal{L}$ be an invariant subspace of an orthogonal transformation $\varphi$ . Prove that $\mathcal{L}^{\perp }$ is also invariant under $\varphi$ . How is this result related to problem $25. 55?$

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Problem 29.50

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

An orthogonal transformation is given in an orthonormal basis by the matrix $A$ . Find the transition matrix $S$ to the canonical basis and the matrix $A^{\prime }$ of the transformation in this basis:

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(1)

$\frac{1}{\sqrt{2}}\left[\begin{smallmatrix} 1 & 1 \\ 1 & - 1\end{smallmatrix}\right]$

(2)

$\frac{1}{4}\left[\begin{smallmatrix} - 1 & - 3 & - \sqrt{6} \\ - 3 & - 1 & \sqrt{6} \\ \sqrt{6} & - \sqrt{6} & 2\end{smallmatrix}\right]$

(3)

$\mathbf{A}_{414}$ . $\mathbf{A}_{414} = \left[\begin{smallmatrix} 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{smallmatrix}\right]$ .

(4)

$\left[\begin{smallmatrix} 0 & 0 & 0 & 0 & 0 & - 1 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0\end{smallmatrix}\right]$ .

Problem 29.51

Prove that the transformation $\varphi^{ * } \varphi$ is orthogonal if and only if $\varphi$ is orthogonal.

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