Math Problems Vault

Problem II.3.1 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section II.3 "Projections and Idempotents; Invariant and Reducing Subspaces"

Let $\mathcal{H}$ be the two-dimensional real Hilbert space $\mathbb {R}^{2}$ , let $\mathcal{M} \equiv \left\{ (x,0) \in \mathbb {R}^{2}: x \in \mathbb {R}\right\}$ and let $\mathcal{N} \equiv \left\{ (x, x \tan \theta ): x \in \mathbb {R}\right\}$ , where $0 < \theta < \frac{1}{2}\pi$ . Find a formula for the idempotent $E_{\theta }$ with $\operatorname {ran} E_{\theta } = \mathcal{M}$ and $\ker E_{\theta } = \mathcal{N}$ . Show that $\left\| E_{\theta }\right\| = (\sin \theta )^{-1}$ .

?

Problem 17.18 from Harville D.A. "Matrix Algebra: Exercises and Solutions" [Har01]More problems in Chapter 17 "Intersections and Sums of Subspaces"

(a)(b)

Let $\mathcal{U}$ and $\mathcal{W}$ represent essentially disjoint subspaces (of $\mathbb {R}^{n \times 1}$ ) whose sum is $\mathbb {R}^{n \times 1}$ , and let $\mathbf{U}$ represent any $n \times s$ matrix such that $C(\mathbf{U}) = \mathcal{U}$ and $\mathbf{W}$ any $n \times t$ matrix such that $C(\mathbf{W}) = \mathcal{W}$ .

?

(a)

Show that the $n \times (s+t)$ partitioned matrix $(\mathbf{U}, \mathbf{W})$ has a right inverse.

(b)

Taking $\mathbf{R}$ to be an arbitrary right inverse of $(\mathbf{U}, \mathbf{W})$ and partitioning $\mathbf{R}$ as $\mathbf{R} = \begin{pmatrix} \mathbf{R}_{1} \\ \mathbf{R}_{2} \end{pmatrix}$ (where $\mathbf{R}_{1}$ has $s$ rows), show that the projection matrix for $\mathcal{U}$ along $\mathcal{W}$ equals $\mathbf{U}\mathbf{R}_{1}$ and that the projection matrix for $\mathcal{W}$ along $\mathcal{U}$ equals $\mathbf{W}\mathbf{R}_{2}$ .

Problem 19.5 from Harville D.A. "Matrix Algebra: Exercises and Solutions" [Har01]More problems in Chapter 19 "Minimization of a Second-Degree Polynomial (in n Variables) Subject to Linear Constraints"

(a)(b)

Let $\mathbf{V}$ represent an $n \times n$ symmetric nonnegative definite matrix and $\mathbf{X}$ an $n \times p$ matrix. Further, let $\mathbf{Z}$ represent any matrix whose columns span $N(\mathbf{X}')$ or, equivalently, $C^{\perp }(\mathbf{X})$ . And, adopt the same terminology as in Exercise 17.20.

?

(a)

Using the result of Part (a) of Exercise 17.20 (or otherwise) show that an $n \times n$ matrix $\mathbf{H}$ is a projection matrix for $C(\mathbf{X})$ along $C(\mathbf{V}\mathbf{Z})$ if and only if $\mathbf{H}'$ is the first ( $n \times n$ ) part of a solution to the consistent linear system

\begin{pmatrix} \mathbf{V} & \mathbf{X} \\ \mathbf{X}' & \mathbf{0} \end{pmatrix} \begin{pmatrix} \mathbf{A} \\ \mathbf{R} \end{pmatrix} = \begin{pmatrix} \mathbf{0} \\ \mathbf{X}' \end{pmatrix} \tag {E.1}

(in an $n \times n$ matrix $\mathbf{A}$ and a $p \times n$ matrix $\mathbf{R}$ ).

(b)

Letting $\mathbf{U}$ represent any $p \times p$ matrix such that $C(\mathbf{X}) \subset C(\mathbf{V} + \mathbf{X}\mathbf{U}\mathbf{X}')$ and letting $\mathbf{W}$ represent an arbitrary generalized inverse of $\mathbf{V} + \mathbf{X}\mathbf{U}\mathbf{X}'$ , show that an $n \times n$ matrix $\mathbf{H}$ is a projection matrix for $C(\mathbf{X})$ along $C(\mathbf{V}\mathbf{Z})$ if and only if

\mathbf{H} = \mathbf{P}_{\mathbf{X},\mathbf{W}} + \mathbf{K}[\mathbf{I} - (\mathbf{V} + \mathbf{X}\mathbf{U}\mathbf{X}')\mathbf{W}]

for some $n \times n$ matrix $\mathbf{K}$ .

Problem 21.6 from Беклемишева Л.А., Беклемишев Д.В., Петрович А.Ю., Чубаров И.А. "Сборник задач по аналитической геометрии и линейной алгебре" [ББП+23]More problems in Section 21 "Sum and intersection of subspaces"

Find the projection of the given vector $\mathbf{x}$ of the $n$ -dimensional arithmetic space onto the vector subspace $\mathcal{P}$ parallel to the vector subspace $\mathcal{Q}$ , where $\mathcal{P}$ is the linear span of the system of vectors $\mathbf{a}_{1}, \ldots , \mathbf{a}_{k}$ , and $\mathcal{Q}$ is the linear span of the system of vectors $\mathbf{b}_{1}, \ldots , \mathbf{b}_{l}$ :

?

(1)

$n = 2, \quad \mathbf{x} = \mathbf{c}_{32}, \quad \mathbf{a}_{1} = \mathbf{c}_{30}, \quad \mathbf{b}_{1} = \mathbf{c}_{34}$ . $\mathbf{c}_{32} = \left[\begin{smallmatrix} 2 \\ -2 \end{smallmatrix}\right], \mathbf{c}_{30} = \left[\begin{smallmatrix} -1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{34} = \left[\begin{smallmatrix} 1 \\ 3 \end{smallmatrix}\right]$ .

(2)

$n = 2, \quad \mathbf{x} = \mathbf{c}_{37}, \quad \mathbf{a}_{1} = \mathbf{c}_{30}, \quad \mathbf{b}_{1} = \mathbf{c}_{34}$ . $\mathbf{c}_{37} = \left[\begin{smallmatrix} 2 \\ 6 \end{smallmatrix}\right], \mathbf{c}_{30} = \left[\begin{smallmatrix} -1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{34} = \left[\begin{smallmatrix} 1 \\ 3 \end{smallmatrix}\right]$ .

(3)

$n = 2, \quad \mathbf{x} = \mathbf{c}_{35}, \quad \mathbf{a}_{1} = \mathbf{c}_{30}, \quad \mathbf{b}_{1} = \mathbf{c}_{34}$ . $\mathbf{c}_{35} = \left[\begin{smallmatrix} 1 \\ 4 \end{smallmatrix}\right], \mathbf{c}_{30} = \left[\begin{smallmatrix} -1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{34} = \left[\begin{smallmatrix} 1 \\ 3 \end{smallmatrix}\right]$ .

(4)

$n = 3, \mathbf{x} = \mathbf{c}_{146}, \mathbf{a}_{1} = \mathbf{c}_{66}, \mathbf{a}_{2} = \mathbf{c}_{121}, \mathbf{a}_{3} = \mathbf{c}_{122}, \mathbf{b}_{1} = \mathbf{c}_{145}$ . $\mathbf{c}_{146} = \left[\begin{smallmatrix} 1 \\ 4 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{66} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{121} = \left[\begin{smallmatrix} -3 \\ 2 \\ 0 \end{smallmatrix}\right], \mathbf{c}_{122} = \left[\begin{smallmatrix} -2 \\ 3 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{145} = \left[\begin{smallmatrix} 2 \\ 0 \\ -1 \end{smallmatrix}\right]$ .

(5)

$n = 4, \mathbf{x} = \mathbf{c}_{201}, \mathbf{a}_{1} = \mathbf{c}_{166}, \mathbf{a}_{2} = \mathbf{c}_{199}, \mathbf{b}_{1} = \mathbf{c}_{197}, \mathbf{b}_{2} = \mathbf{c}_{198}$ . $\mathbf{c}_{201} = \left[\begin{smallmatrix} 1 \\ -7 \\ 5 \\ -2 \end{smallmatrix}\right], \mathbf{c}_{166} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{199} = \left[\begin{smallmatrix} 3 \\ -5 \\ 7 \\ 2 \end{smallmatrix}\right], \mathbf{c}_{197} = \left[\begin{smallmatrix} 1 \\ 1 \\ 2 \\ 2 \end{smallmatrix}\right], \mathbf{c}_{198} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 3 \end{smallmatrix}\right]$ .

Problem 22.10 from Беклемишева Л.А., Беклемишев Д.В., Петрович А.Ю., Чубаров И.А. "Сборник задач по аналитической геометрии и линейной алгебре" [ББП+23]More problems in Section 22 "Complex linear spaces"

(1)(2)(3)

Find the projection of the given vector $\mathbf{x}$ of the two-dimensional complex arithmetic space onto the vector subspace $\mathcal{P}$ parallel to the vector subspace $\mathcal{Q}$ , where $\mathcal{P}$ is spanned by the vector $\mathbf{c}_{44}$ , and $\mathcal{Q}$ is spanned by the vector $\mathbf{c}_{40}$ . $\mathbf{c}_{44} = \left[\begin{smallmatrix} i \\ 2 \end{smallmatrix}\right], \mathbf{c}_{40} = \left[\begin{smallmatrix} 1+i \\ 3 \end{smallmatrix}\right]$ .

?

(1)

$\mathbf{x} = \mathbf{c}_{26}$ . $\mathbf{c}_{26} = \left[\begin{smallmatrix} -1 \\ 2i \end{smallmatrix}\right]$ .

(2)

$\mathbf{x} = \mathbf{c}_{42}$ . $\mathbf{c}_{42} = \left[\begin{smallmatrix} 3-i \\ 3-6 i \end{smallmatrix}\right]$ .

(3)

$\mathbf{x} = \mathbf{c}_{45}$ . $\mathbf{c}_{45} = \left[\begin{smallmatrix} 2+3 i \\ -3 i \end{smallmatrix}\right]$ .

Problem 23.8 from Беклемишева Л.А., Беклемишев Д.В., Петрович А.Ю., Чубаров И.А. "Сборник задач по аналитической геометрии и линейной алгебре" [ББП+23]More problems in Subsection 23.1 "Examples of linear mappings and transformations. Kernel, range. Matrices of linear mappings and transformations (23.1-23.51)"

Let $\mathbf{a}$ and $\mathbf{n}$ be nonzero vectors of a three-dimensional geometric vector space, with $(\mathbf{a}, \mathbf{n}) \neq 0, \mathcal{L}_{1}$ a line with directing vector $\mathbf{a}$ , and $\mathcal{L}_{2}$ a plane with normal vector $\mathbf{n}$ . Write a formula for the transformation $\varphi$ , verify its linearity, find the kernel, the range, and the rank, if $\varphi$ is:

?

(1)

orthogonal projection onto $\mathcal{L}_{2}$

(2)

orthogonal projection onto $\mathcal{L}_{1}$

(3)

projection onto $\mathcal{L}_{2}$ parallel to the vector $\mathbf{a}$

(4)

projection onto $\mathcal{L}_{1}$ parallel to $\mathcal{L}_{2}$

(5)

orthogonal reflection with respect to $\mathcal{L}_{2}$

(6)

orthogonal reflection with respect to $\mathcal{L}_{1}$

(7)

reflection in $\mathcal{L}_{2}$ parallel to the vector $\mathbf{a}$

(8)

reflection in $\mathcal{L}_{1}$ parallel to $\mathcal{L}_{2}$ .

Problem 23.9 from Беклемишева Л.А., Беклемишев Д.В., Петрович А.Ю., Чубаров И.А. "Сборник задач по аналитической геометрии и линейной алгебре" [ББП+23]More problems in Subsection 23.1 "Examples of linear mappings and transformations. Kernel, range. Matrices of linear mappings and transformations (23.1-23.51)"

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

Compute the matrix of the orthogonal projection of the space $\mathcal{E}_{3}$ onto the subspace $\mathcal{L}$ , if $\mathcal{L}$ is:

?

(1)

the line $x = z = 0$

(2)

the line $x = y = z$

(3)

the plane $x + y + z = 0$

(4)

the plane spanned by the vectors $\mathbf{a}(- 1, 1, - 1)$ and $\mathbf{b}(1, - 3, 2)$ .

In this problem, the vector subspaces of the three-dimensional geometric vector space $\mathcal{E}_{3}$ are given by their equations in some orthonormal basis.

Problem 23.10 from Беклемишева Л.А., Беклемишев Д.В., Петрович А.Ю., Чубаров И.А. "Сборник задач по аналитической геометрии и линейной алгебре" [ББП+23]More problems in Subsection 23.1 "Examples of linear mappings and transformations. Kernel, range. Matrices of linear mappings and transformations (23.1-23.51)"

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

Compute the matrix of the projection of the space $\mathcal{E}_{3}$ onto the subspace $\mathcal{L}$ parallel to the subspace $\mathcal{M}$ , if:

?

(1)

$\mathcal{L}$ is defined by the equation $x = 0$ , $\mathcal{M}$ by the equations $2x = 2y = - z$

(2)

$\mathcal{L}$ has the equation $x = y$ , $\mathcal{M}$ is defined by the system of equations $x + y + z = 0, 2x + y + 4z = 0$

(3)

$\mathcal{L}$ is defined by the equations $- 20x = 15y = 12z$ , $\mathcal{M}$ by the equation $2x + 3y - z = 0$

(4)

$\mathcal{L}$ is defined by the system of equations $x - y + z = 0, 2x - 3y + 4z = 0$ , $\mathcal{M}$ by the equation $2x + 3y - 4z = 0$ .

In this problem, the vector subspaces of the three-dimensional geometric vector space $\mathcal{E}_{3}$ are given by their equations in some orthonormal basis.

Problem 23.11 from Беклемишева Л.А., Беклемишев Д.В., Петрович А.Ю., Чубаров И.А. "Сборник задач по аналитической геометрии и линейной алгебре" [ББП+23]More problems in Subsection 23.1 "Examples of linear mappings and transformations. Kernel, range. Matrices of linear mappings and transformations (23.1-23.51)"

The transformations of the space $\mathcal{E}_{3}$ from problems 23.9 (items 1-4, orthogonal projection) and 23.10 (items 5-8, projection parallel to $\mathcal{M}$ ) are considered as linear maps of the space $\mathcal{E}_{3}$ onto the subspace $\mathcal{L}$ . Compute the matrix of each of these maps in some pair of bases.

?

(1)

$\mathcal{L}$ is the line $x = z = 0$

(2)

$\mathcal{L}$ is the line $x = y = z$

(3)

$\mathcal{L}$ is the plane $x + y + z = 0$

(4)

$\mathcal{L}$ is the plane spanned by the vectors $\mathbf{a}(- 1, 1, - 1)$ and $\mathbf{b}(1, - 3, 2)$

(5)

$\mathcal{L}$ is defined by the equation $x = 0$ , $\mathcal{M}$ by the equations $2x = 2y = - z$

(6)

$\mathcal{L}$ has the equation $x = y$ , $\mathcal{M}$ is defined by the system of equations $x + y + z = 0, 2x + y + 4z = 0$

(7)

$\mathcal{L}$ is defined by the equations $- 20x = 15y = 12z$ , $\mathcal{M}$ by the equation $2x + 3y - z = 0$

(8)

$\mathcal{L}$ is defined by the system of equations $x - y + z = 0, 2x - 3y + 4z = 0$ , $\mathcal{M}$ by the equation $2x + 3y - 4z = 0$ .

In this problem, the vector subspaces of the three-dimensional geometric vector space $\mathcal{E}_{3}$ are given by their equations in some orthonormal basis.

Problem 23.65 from Беклемишева Л.А., Беклемишев Д.В., Петрович А.Ю., Чубаров И.А. "Сборник задач по аналитической геометрии и линейной алгебре" [ББП+23]More problems in Subsection 23.2 "Matrices of linear mappings and transformations in different bases. Similar matrices (23.52-23.77)"

(1)(2)(3)

Compute the matrix of the linear transformation $\varphi$ of the set of vectors of the plane with a given basis on it, if $\varphi$ is:

?

(1)

reflection of the plane in the line $x + 2y = 0$ parallel to the line $x + 3y = 0$

(2)

projection of the plane onto the line $x + y = 0$ parallel to the line $4x + 5y = 0$

(3)

contraction with coefficient $\lambda = 2$ toward the line $3x - 2y = 0$ parallel to the line $x - y = 0$ .

Problem 23.66 from Беклемишева Л.А., Беклемишев Д.В., Петрович А.Ю., Чубаров И.А. "Сборник задач по аналитической геометрии и линейной алгебре" [ББП+23]More problems in Subsection 23.2 "Matrices of linear mappings and transformations in different bases. Similar matrices (23.52-23.77)"

Compute the matrix of the linear transformation $\varphi$ of the three-dimensional geometric vector space (in which an orthonormal basis is given), using a suitable change of basis, if $\varphi$ is:

?

(1)

projection onto the plane $3x - y = 0$ parallel to the line $x + z = 0, x + y + 2z = 0$

(2)

reflection of the space in the line $x = y = - 2z$ parallel to the plane $x + y + 3z = 0$

(3)

contraction with coefficient $\lambda = 2$ toward the plane $x - 2z = 0$ parallel to the line $x = y = z$

(4)

rotation about the line $x = y = - z$ by angle $\pi / 2$

(5)

rotation about the line $x + y = 0, y - z = 0$ by an angle such that the first of the given planes goes into the second.

Problem 28.1 from Беклемишева Л.А., Беклемишев Д.В., Петрович А.Ю., Чубаров И.А. "Сборник задач по аналитической геометрии и линейной алгебре" [ББП+23]More problems in Subsection 28.1 "Examples of linear transformations of Euclidean space (28.1-28.9)"

(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

?

(1)

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

(2)

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

Problem 28.2 from Беклемишева Л.А., Беклемишев Д.В., Петрович А.Ю., Чубаров И.А. "Сборник задач по аналитической геометрии и линейной алгебре" [ББП+23]More problems in Subsection 28.1 "Examples of linear transformations of Euclidean space (28.1-28.9)"

(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:

?

(1)

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

(2)

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

Problem 28.3 from Беклемишева Л.А., Беклемишев Д.В., Петрович А.Ю., Чубаров И.А. "Сборник задач по аналитической геометрии и линейной алгебре" [ББП+23]More problems in Subsection 28.1 "Examples of linear transformations of Euclidean space (28.1-28.9)"

(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:

?

(1)

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

(2)

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

Problem 30.1 from Беклемишева Л.А., Беклемишев Д.В., Петрович А.Ю., Чубаров И.А. "Сборник задач по аналитической геометрии и линейной алгебре" [ББП+23]More problems in Subsection 30.1 "Linear transformations of a unitary space. Examples of transformations. Adjoint transformation (30.1-30.12)"

(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:

?

(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 from Беклемишева Л.А., Беклемишев Д.В., Петрович А.Ю., Чубаров И.А. "Сборник задач по аналитической геометрии и линейной алгебре" [ББП+23]More problems in Subsection 30.1 "Linear transformations of a unitary space. Examples of transformations. Adjoint transformation (30.1-30.12)"

(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 $\mathcal{L}$ :

?

(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 39.17 from Кострикин А.И. "Сборник задач по алгебре" [Кос15]More problems in Section 39 "Definition of a linear operator. The image, the kernel, the matrix of a linear operator"

Let a vector space $V$ be the direct sum of subspaces $L_{1}$ and $L_{2}$ with bases $\left(a_{1}, \ldots , a_{k}\right)$ and $\left(a_{k+1}, \ldots , a_{n}\right)$ . Prove that the projection onto $L_{1}$ parallel to $L_{2}$ is a linear operator, and find its matrix in the basis $\left(a_{1}, \ldots , a_{n}\right)$ .

?

Problem 5.13.18 from Meyer C.D. "Matrix Analysis and Applied Linear Algebra" [Mey00]More problems in Section 5.13 "Orthogonal Projection"

(a)(b)(c)

For $u,w \in \mathbb {R}^{n}$ such that $u^{T}w \neq 0$ , let $M = u^{\perp }$ and $W = \left\langle w\right\rangle$ .

?

(a)

Explain why $\mathbb {R}^{n} = M\oplus W$ .

(b)

For $b \in \mathbb {R}^{n}$ , explain why the oblique projection of $b$ onto $M$ along $W$ is given by $p = b - \dfrac {u^{T}b}{u^{T}w}w$ .

(c)

For a given scalar $\beta$ , let $H$ be the hyperplane $H=\left\{ x \mid u^{T}x=\beta \right\}$ . Explain why the oblique projection of $b$ onto $H$ along $W$ should be given by $p = b - \left(\dfrac {u^{T}b-\beta }{u^{T}w}\right)w$ .

Problem 5.13.20 from Meyer C.D. "Matrix Analysis and Applied Linear Algebra" [Mey00]More problems in Section 5.13 "Orthogonal Projection"

Assume a nonsingular system $A_{n\times n}x=b$ has been row scaled so that $\left\| A_{i*}\right\|_{2}=1$ for each $i$ , and let $H_{i} = \left\{ x \mid A_{i*}x=b_{i}\right\}$ . In theory, the system can be solved by $n-1$ oblique projections (Exercise 5.13.18): an arbitrary point $p_{1}$ in $H_{1}$ is projected obliquely onto $H_{2}$ along $H_{1}$ to produce $p_{2} \in H_{1}\cap H_{2}$ , then $p_{2}$ is projected onto $H_{3}$ along $H_{1}\cap H_{2}$ to produce $p_{3}$ , and so on. Since $\bigcap_{i=1}^{k} H_{i}$ is generally unknown, the procedure is modified — use the following figure, showing the $n=3$ case, as a guide.

$Successive oblique projections for n=3: p_1^{(1)} is projected through p_2^{(1)}, p_3^{(1)} onto \mathcal{H}_2, producing points in \mathcal{H}_1 \cap \mathcal{H}_2.$ Successive oblique projections for n=3: p_1^{(1)} is projected through p_2^{(1)}, p_3^{(1)} onto \mathcal{H}_2, producing points in \mathcal{H}_1 \cap \mathcal{H}_2.

Step 0. Begin with any set $\left\{ p_{1}^{(1)}, p_{2}^{(1)}, \ldots , p_{n}^{(1)}\right\} \subset H_{1}$ such that $\left\{ p_{1}^{(1)}-p_{2}^{(1)}, \ldots , p_{1}^{(1)}-p_{n}^{(1)}\right\}$ is linearly independent and $A_{2*}(p_{1}^{(1)}-p_{k}^{(1)}) \neq 0$ for $k=2,\ldots ,n$ . Step 1. In turn, project $p_{1}^{(1)}$ onto $H_{2}$ through $p_{2}^{(1)},\ldots ,p_{n}^{(1)}$ to produce $\left\{ p_{2}^{(2)},\ldots ,p_{n}^{(2)}\right\} \subset H_{1}\cap H_{2}$ . Step 2. Project $p_{2}^{(2)}$ onto $H_{3}$ through $p_{3}^{(2)},\ldots ,p_{n}^{(2)}$ to produce $\left\{ p_{3}^{(3)},\ldots ,p_{n}^{(3)}\right\} \subset H_{1}\cap H_{2}\cap H_{3}$ , and so on. Step $n-1$ . Project $p_{n-1}^{(n-1)}$ through $p_{n}^{(n-1)}$ to produce $p_{n}^{(n)} \in \bigcap_{i=1}^{n} H_{i}$ . Then $x = p_{n}^{(n)}$ is the solution. Explain why the following algorithm performs the computations of Steps $1,2,\ldots ,n-1$ . For $i=2$ to $n$ For $j=i$ to $n$ $x_{j} \leftarrow x_{j} - (A_{i*}x_{i-1}-b_{i})\dfrac {x_{i-1}-x_{j}}{A_{i*}(x_{i-1}-x_{j})}$ $x \leftarrow x_{n}$ (the solution of the system)

?

Problem 5.9.1 from Meyer C.D. "Matrix Analysis and Applied Linear Algebra" [Mey00]More problems in Section 5.9 "Complementary Subspaces"

Let $X$ and $Y$ be subspaces of $\mathbb {R}^{3}$ whose respective bases are $B_{X} = \left\{ \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}\right\}$ and $B_{Y} = \left\{ \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}\right\}$ .

?

(a)

Explain why $X$ and $Y$ are complementary subspaces of $\mathbb {R}^{3}$ .

(b)

Determine the projector $P$ onto $X$ along $Y$ as well as the complementary projector $Q$ onto $Y$ along $X$ .

(c)

Determine the projection of $v = \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}$ onto $Y$ along $X$ .

(d)

Verify that $P$ and $Q$ are both idempotent.

(e)

Verify that $\mathcal{R}(P) = X = \mathcal{N}(Q)$ and $\mathcal{N}(P) = Y = \mathcal{R}(Q)$ .

Remark.

?

(5.9.4): $X, Y$ (with bases $B_{X},B_{Y}$ ) are complementary subspaces of $V$ if and only if $B_{X}\cap B_{Y}=\emptyset$ and $B_{X}\cup B_{Y}$ is a basis for $V$ . (5.9.12): if the columns of $X,Y$ (matrices) are bases for $X,Y$ , the projector onto $X$ along $Y$ is $P = \left[X\mid 0\right]\left[X\mid Y\right]^{-1}$ .

Problem 5.9.4 from Meyer C.D. "Matrix Analysis and Applied Linear Algebra" [Mey00]More problems in Section 5.9 "Complementary Subspaces"

Explain why $\mathbb {R}^{n \times n} = S \oplus K$ , where $S$ and $K$ are the subspaces of $n\times n$ symmetric and skew-symmetric matrices, respectively. What is the projection of $A = \begin{pmatrix} 1& 2& 3 \\ 4& 5& 6 \\ 7& 8& 9 \end{pmatrix}$ onto $S$ along $K$ ?

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Remark.

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Exercise 3.2.6 shows that every $A \in \mathbb {R}^{n\times n}$ has the unique decomposition $A = \frac{A+A^{T}}{2} + \frac{A-A^{T}}{2}$ , a symmetric matrix plus a skew-symmetric matrix.

Problem 5.9.11 from Meyer C.D. "Matrix Analysis and Applied Linear Algebra" [Mey00]More problems in Section 5.9 "Complementary Subspaces"

Suppose that $X$ and $Y$ are complementary subspaces of $\mathbb {R}^{n}$ , and let $B = \left[X \mid Y\right]$ be a nonsingular matrix in which the columns of $X$ and $Y$ constitute respective bases for $X$ and $Y$ . For an arbitrary vector $v \in \mathbb {R}^{n}$ , explain why the projection of $v$ onto $X$ along $Y$ can be obtained by the following two-step process.

Solve the system $Bz=v$ for $z$ .
Partition $z$ as $z = \begin{pmatrix} z_{1} \\ z_{2} \end{pmatrix}$ , and set $p = Xz_{1}$ .

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Computing projections along a complementary subspace

References