Math Problems Vault

Problem 30.1

(1)(2)

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

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(1)

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

(2)

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

Problem 30.2

(1)(2)(3)

Let $\mathcal{L} \subset \mathcal{U}$ be the linear span of vectors $a_{1}$ and $a_{2}$ . In an orthonormal basis $e$ of the space $\mathcal{U}$ the coordinate columns of these vectors are given. Write, in the basis $e$ , the matrix $P$ of orthogonal projection onto the subspace $\mathcal{L}$ and the matrix $Q$ of reflection in the subspace $\mathcal{L}$ :

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(1)

$\left[\begin{smallmatrix} 1 \\ i \\ 1 + i\end{smallmatrix}\right], \quad \left[\begin{smallmatrix} i \\ 1 \\ 1-i\end{smallmatrix}\right]$ ;

(2)

$\left[\begin{smallmatrix} i \\ i \\ 1\end{smallmatrix}\right], \quad \left[\begin{smallmatrix} 2i \\ 0 \\ 3\end{smallmatrix}\right]$ ;

(3)

$\left[\begin{smallmatrix} i \\ 1 \\ 1 \\ i\end{smallmatrix}\right], \quad \left[\begin{smallmatrix} 2i \\ 0 \\ 3 \\ i\end{smallmatrix}\right]$ .

Problem 30.3

A vector $a$ is given, and the subspace $\mathcal{L} \subset \mathcal{U}$ 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.

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Problem 30.4

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

In an orthonormal basis, the coordinate column a of the vector $a$ is given. Let $\varphi$ be the reflection in the subspace $\mathcal{L} \subset \mathcal{U}$ defined by the equation $(a, x) = 0$ . Express the matrix of the transformation $\varphi$ in terms of a, if:

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(1)

$\mathbf{a} = \left[\begin{smallmatrix} 3 \\ -1 \\ 2i \end{smallmatrix}\right]$

(2)

$\mathbf{a} = \mathbf{c}_{153}$ . $\mathbf{c}_{153} = \left[\begin{smallmatrix} -i \\ -1 \\ -3 i \end{smallmatrix}\right]$ .

(3)

$\mathbf{a} = \mathbf{c}_{131}$ . $\mathbf{c}_{131} = \left[\begin{smallmatrix} 1 \\ -i \\ 1+i \end{smallmatrix}\right]$ .

(4)

$\mathbf{a} = \mathbf{c}_{215}$ . $\mathbf{c}_{215} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 1 - i \end{smallmatrix}\right]$ .

Problem 30.5

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

Prove that in a unitary space the operation of passing to the adjoint transformation has the following properties:

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$(\varphi + \psi )^{ * } = \varphi^{ * } + \psi^{ * }$ ;

(2)

$(\varphi \psi )^{ * } = \psi^{ * } \varphi^{ * }$ ;

(3)

$(\alpha \varphi )^{ * } = \bar{\alpha } \varphi^{ * }$ ;

(4)

if $\varphi$ has an inverse, then $\varphi^{ * }$ also has an inverse, and $\left(\varphi^{ * }\right)^{ - 1} = \left(\varphi^{ - 1}\right)^{ * }$ .

Problem 30.6

(1)(2)(3)

In an orthonormal basis, the matrix of a linear transformation of a unitary space is given. Find the matrix of the adjoint transformation:

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$\mathbf{A}_{88}$ . $\mathbf{A}_{88} = \left[\begin{smallmatrix} 1 & i \\ i & 1 \end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{103}$ . $\mathbf{A}_{103} = \left[\begin{smallmatrix} 0 & i \\ -i & 0 \end{smallmatrix}\right]$ .

(3)

$\mathbf{A}_{100}$ . $\mathbf{A}_{100} = \left[\begin{smallmatrix} 1 & 2 i \\ i & -1 \end{smallmatrix}\right]$ .

Problem 30.7

(1)(2)(3)

Let $A$ be the matrix of a linear transformation in a basis with Gram matrix $\Gamma$ . Find the matrix of the adjoint transformation:

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(1)

$A = \mathbf{A}_{92}, \quad \Gamma = \mathbf{A}_{87}$ . $\mathbf{A}_{92} = \left[\begin{smallmatrix} i & 1 \\ 1 & i \end{smallmatrix}\right], \mathbf{A}_{87} = \left[\begin{smallmatrix} 1 & 1+i \\ 1-i & 3 \end{smallmatrix}\right]$ .

(2)

$A = \mathbf{A}_{97}, \quad \Gamma = \mathbf{A}_{86}$ . $\mathbf{A}_{97} = \left[\begin{smallmatrix} i & 1 \\ -1 & 2 i \end{smallmatrix}\right], \mathbf{A}_{86} = \left[\begin{smallmatrix} 5 & i \\ -i & 1 \end{smallmatrix}\right]$ .

(3)

$A = \mathbf{A}_{375}, \quad \Gamma = \mathbf{A}_{388}$ . $\mathbf{A}_{375} = \left[\begin{smallmatrix} 1 & 1+i & 1-i \\ 1 & 1 & 1 \\ -1 & -i & i \end{smallmatrix}\right], \mathbf{A}_{388} = \left[\begin{smallmatrix} 2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 1 \end{smallmatrix}\right]$ .

Problem 30.8

(1)(2)

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(1)

Let a vector $x$ be an eigenvector for the transformation $\varphi$ with eigenvalue $\lambda$ and an eigenvector for $\varphi^{ * }$ with eigenvalue $\mu$ . Prove that $\lambda = \bar{\mu }$ .

(2)

Prove that the transformation $\varphi^{ * }$ , adjoint to a transformation $\varphi$ with eigenvalues $\lambda_{1}, \ldots , \lambda_{n}$ , has eigenvalues $\overline{\lambda_{1}}, \ldots , \overline{\lambda_{n}}$ .

Problem 30.9

Let the subspace $\mathcal{L}$ be invariant under the transformation $\varphi$ . Prove that $\mathcal{L}^{\perp }$ is invariant under the adjoint transformation $\varphi^{ * }$ .

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Problem 30.10

(1)(2)

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(1)

Prove that the set of values of a linear transformation $\varphi$ of a unitary space coincides with the orthogonal complement of the kernel of the adjoint transformation $\varphi^{ * }$ .

(2)

Prove that the Fredholm theorem for complex systems of linear equations also holds in the following formulation: the system $A\mathbf{x} = \mathbf{b}$ is consistent if and only if every solution of the system of linear equations $\bar{A}^{T} \mathbf{y} = \mathbf{0}$ satisfies the equality $\mathbf{y}^{T} \bar{b} = 0$ .

Problem 30.11

(1)(2)

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(1)

Prove that every linear transformation of a unitary space has an $(n - 1)$ -dimensional invariant subspace. Is this statement true for complex vector spaces?

(2)

Prove that for every linear transformation of a unitary space there exists an orthonormal basis in which the matrix of the transformation is upper triangular.

Problem 30.12

(1)(2)(3)

In an orthonormal basis, the matrix $A$ of a linear transformation $\varphi$ is given. Find the transition matrix $S$ to an orthonormal basis in which $\varphi$ has a triangular matrix $A^{\prime }$ , and write the matrix $A^{\prime }$ :

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(1)

$A =\left[\begin{smallmatrix} i & 1 - i & 0 \\ 0 & 1 & 0 \\ 2 & 1 + i & 2 - i\end{smallmatrix}\right]$

(2)

$A =\left[\begin{smallmatrix} - 1 & - 1 + i & - 1 - i \\ 1 + i & 1 & 1 + i \\ 1 - i & 1 - i & 1\end{smallmatrix}\right]$

(3)

$A =\left[\begin{smallmatrix} i & i & - i \\ i & 1 & 0 \\ 1 & 1 + i & - i\end{smallmatrix}\right]$