Chapter VI

Denote by $E_{i j}$ the $m \times n$ matrix whose entry at the intersection of the $i$-th row and the $j$-th column equals 1 , and all the remaining entries are zero. Verify that these $mn$ matrices form a basis in the linear space of $m \times n$ matrices. (This basis is called the standard basis of the given space.) What are the coordinates of a matrix $A$ with entries $a_{i j}$ in the standard basis?

Prove that the upper triangular matrices of order $n$ form a linear space with respect to the usual operations on matrices. Find the dimension of this space and some basis in it.

In the linear space of polynomials of degree $\leqslant 3$ in the variable $t$ two bases are given: $1, t, t^{2}, t^{3}$ and $1, t-a,(t-a)^{2}$, $(t-a)^{3}$. Find the change-of-basis matrix from the first basis to the second and, using it, the expansion of a polynomial $p(t)$ in the second basis.

How are two bases $e_{1}, \ldots , e_{n}$ and $f_{1}, \ldots , f_{n}$ positioned relative to each other if the change-of-basis matrix from $\mathbf{e}$ to $\mathbf{f}$ is upper triangular? Prove from these considerations that the inverse of an upper triangular matrix is also upper triangular.

How are two bases oriented relative to each other if $f_{1}=e_{1}+e_{2} ; f_{2}=e_{2}+e_{3} ; f_{3}=e_{3}+e_{4} ; f_{4}=e_{4}-e_{1}$ ?

In a linear space $\mathcal{L}$ the vectors $a_{1}, a_{2}$ and $a_{3}$ are given with coordinate columns

in the basis $e_{1}, e_{2}, e_{3}, e_{4}$. Find a basis of their linear span $\mathcal{L}^{\prime }$.

Find a system of equations defining the subspace $\mathcal{L}^{\prime }$ from problem 1.

Find some subspace $\mathcal{M}^{\prime }$ which together with the subspace $\mathcal{L}^{\prime }$ from problem 1 satisfies the condition $\mathcal{L}=\mathcal{L}^{\prime } \oplus \mathcal{M}^{\prime }$.

In problem 1, $\mathcal{L}^{\prime }$ is the linear span of the vectors $a_{1}, a_{2}, a_{3}$ of the linear space $\mathcal{L}$ whose coordinate columns in the basis $e_{1}, e_{2}, e_{3}, e_{4}$ are $\| 1,2,3,4\|^{T}$, $\| 5,6,7,8\|^{T}$ and $\| 9,10,11,12\|^{T}$.

The subspace $\mathcal{L}^{\prime }$ is defined in problem 1: it is the linear span of the vectors $a_{1}, a_{2}, a_{3}$ of a linear space $\mathcal{L}$ whose coordinate columns in the basis $e_{1}, e_{2}, e_{3}, e_{4}$ are $\| 1,2,3,4\|^{T}$, $\| 5,6,7,8\|^{T}$ and $\| 9,10,11,12\|^{T}$. The subspace $\mathcal{L}^{\prime \prime }$ is spanned by the vectors $b_{1}$ and $b_{2}$ with coordinates 1, 1, 1, 2 and 2, 2, 2, 5. Find

In a four-dimensional space there are given

All square matrices of order 2 are multiplied on the right by the matrix

This defines a map $\mathrm{A}$ of the space of matrices of order 2 into the space of matrices of sizes 2 × 3.

What condition must the matrix $C$ of sizes 2 × 3 satisfy in order for the map $\mathrm{A}$, defined in problem 1, to be injective? Can it be surjective?

Let $\mathcal{C}^{k}$ be the linear space of functions having $k>1$ continuous derivatives on the segment $[0,1]$. Differentiation maps $\mathcal{C}^{k}$ into $\mathcal{C}^{k-1}$. Verify that this is a linear map. Will it be

Let $\mathrm{A}: \mathcal{L} \rightarrow \overline{\mathcal{L}}$ and $\mathcal{M}=\mathrm{A}(\mathcal{L})$. Define a map $\mathrm{A}^{\prime }: \mathcal{L} \rightarrow \mathcal{M}$ by the equality $\mathrm{A}^{\prime }(x)=\mathrm{A}(x)$. Prove that

Let $\mathcal{L}=\mathcal{L}_{1} \oplus \mathcal{L}_{2}$ and $x=x_{1}+x_{2}, x_{1} \in \mathcal{L}_{1}, x_{2} \in \mathcal{L}_{2}$. Define transformations $\mathrm{P}_{1}$ and $\mathrm{P}_{2}$ of the space $\mathcal{L}$ by the formulas: $\mathrm{P}_{1}(x)=x_{1}$ and $\mathrm{P}_{2}(x)=x_{2}$ (such transformations are called projections). Prove that

where $\mathrm{O}$ is the zero transformation and $\mathrm{E}$ is the identity transformation.

For any linear map $\mathrm{A}: \mathcal{L} \rightarrow \overline{\mathcal{L}}$ there exist bases in $\mathcal{L}$ and $\overline{\mathcal{L}}$ such that $A$ will have the matrix

( $E_{r}$ is the identity submatrix of order $r$, the remaining entries, if there are any, equal zero.) Prove this theorem by reducing the matrix of the linear map to the form (4) by elementary row and column transformations.

Let $\mathrm{A}$ be a linear map. Is it true that

Prove that the rank of a product of maps does not exceed the ranks of these maps. (This will be another proof of the estimate for the rank of a product of matrices from § 3 of ch. V: the rank of a product of two matrices does not exceed the ranks of the factors, $\mathrm{Rg} A B \leqslant \mathrm{Rg} A$ and $\mathrm{Rg} A B \leqslant \mathrm{Rg} B$.)

Let $\mathrm{B}$ be injective. Prove that for any $\mathrm{A}$ one has $\mathrm{RgBA}=\mathrm{RgA}$.

Prove that every subspace $\mathcal{L}^{\prime }$ lying in $\operatorname {Ker} \mathrm{A}$, and every subspace $\mathcal{L}^{\prime \prime }$ containing $\operatorname {Im} \mathrm{A}$, are invariant with respect to $\mathrm{A}$.

Prove that the sum and the intersection of invariant subspaces are invariant.

Let $\mathrm{A}$ be a linear transformation of a real space, $\lambda$ a complex root of its characteristic polynomial, $p=(\lambda +\bar{\lambda }) / 2$ and $q=\lambda \bar{\lambda }$. Prove that the dimension of the subspace $\mathcal{L}^{\prime }=\operatorname {Ker}\left(\mathrm{A}^{2}-p A+q E\right)$ is an even number.

Let $\mathrm{A}: \mathcal{L} \rightarrow \mathcal{L}$. Prove that $\mathcal{L}=\operatorname {Ker} \mathrm{A} \oplus \operatorname {Im} \mathrm{A}$ if and only if $\operatorname {Ker} \mathrm{A}^{2}=\operatorname {Ker} \mathrm{A}$.

Let $\mathcal{L}=\operatorname {Ker} \mathrm{A} \oplus \operatorname {Im} \mathrm{A}$. What form does the matrix of the transformation $\mathrm{A}$ have in the basis $\mathbf{e}$, if $e_{1}, \ldots , e_{r} \in \operatorname {Im} \mathrm{~ A}$, and $e_{r+1}, \ldots \ldots , e_{n} \in \operatorname {Ker} A$ ?

In some basis a transformation is given by the matrix

Does there exist a basis in which its matrix is diagonal? If yes, then find this basis.

Find the eigenvalues and the eigenspaces of the transformation given by the matrix

To every square matrix of order $n$ its transposed matrix is assigned. This defines a transformation $\mathrm{T}$ of the space of matrices. Find its eigenvalues and eigenspaces. Prove from these considerations that every matrix is uniquely represented as a sum of a symmetric and a skew-symmetric matrix.

Let $\mathrm{A}$ be a rotation of three-dimensional vector space about some axis through the angle $\alpha$ and let $A$ be the matrix of this rotation in some basis. Express $\alpha$ through the entries of $A$.

Let $\mathrm{A}$ be diagonalizable. Prove that the restriction of $\mathrm{A}$ to any invariant subspace is also diagonalizable.

The transformation $\mathrm{A}$ is given by the matrix

Find the change-of-basis matrix to some basis in which its matrix is upper triangular, and also the matrix of the transformation in this basis.

Let the matrix $A$ be nonsingular. Represent the matrix $A^{-1}$ as a polynomial in $A$.

Let a transformation $\mathrm{A}$ satisfy the relation $\mathrm{A}^{2}=\mathrm{A}$. Prove that $\mathrm{Rg} \mathrm{A}=\operatorname {tr} \mathrm{A}$.

Let a linear transformation $\mathrm{A}$ be given in some basis of an $n$-dimensional space $\mathcal{L}$ by a matrix $A$, and let an $(n-1)$-dimensional subspace $\mathcal{L}^{\prime }$ be given by the equation $\mathbf{u}^{T} \mathbf{x}=0$. Prove that $\mathcal{L}^{\prime }$ is invariant with respect to $\mathrm{A}$ if and only if there exists a factor $\lambda$ such that $A^{T} \mathbf{u}=\lambda \mathbf{u}$.

Can a linear function on a linear space $\mathcal{L}$ satisfy, for all $x \in \mathcal{L}$, the inequality

Let $\vec{a}$ be a fixed vector of the plane. Assign to every vector $\vec{x}$ the area of the oriented parallelogram built on $\vec{x}$ and $\vec{a}$, or 0 if the vectors are collinear. Verify that this function is linear, and find the row of its coefficients in the basis $\vec{e}_{1}, \vec{e}_{2}$, if $\vec{a}=\alpha \vec{e}_{1}+\beta \vec{e}_{2}$.

Let $k$ be a natural number. Assign to every polynomial of degree at most $n$ the value of its $k$-th derivative at the point $a$. Verify that this defines a linear function. Find its coordinate row in the bases

Let $e_{1}, \ldots , e_{n} \in \mathcal{L}$ and $\mathrm{e}^{* 1}, \ldots , \mathrm{e}^{* n} \in \mathcal{L}^{*}$ be a pair of biorthogonal bases. Verify that for any $x \in \mathcal{L}$ and for any $\mathrm{f} \in \mathcal{L}^{*}$ the equalities hold:

The value of a bilinear function $\mathrm{b}$ in some basis is written as a polynomial in the coordinates $\xi^{i}$ and $\eta^{j}$ of the vectors $x$ and $y$ :

Write the matrix of this bilinear function if the space is

How will the matrix of the bilinear function from problem $1(a)$ change if we pass to the basis $e_{1}^{\prime }=e_{1}+e_{2}, e_{2}^{\prime }=e_{2}+e_{3}, e_{3}^{\prime }=e_{3}$ ?

Write the matrix of the quadratic form $\left(\xi^{1}\right)^{2}+\xi^{1} \xi^{2}+\left(\xi^{2}\right)^{2}$.

Reduce to canonical form the quadratic form with matrix

The null space of a symmetric bilinear function $\mathrm{b}$ is the set of vectors $x$ such that $\mathrm{b}(x, y)=0$ for all $y$. Verify that this is a linear subspace. How is its dimension $s$ related to the rank of $\mathrm{b}$? What will the matrix of the function $\mathrm{b}$ be in a basis whose last $s$ vectors lie in the null space?

$B n$-dimensional space $m$ quadratic forms are given. Under what condition does there exist a basis in which they can all be represented as polynomials in the first $k<n$ coordinates of a vector?

Let $A$ be a square matrix of order $n$ and rank $r$. For the quadratic form with matrix $A^{T} A$ determine

A quadratic form with matrix $B$ is positive definite if and only if there exists a nonsingular upper triangular matrix $R$ such that $B=R^{T} R$. Prove this.

A quadratic form $\mathrm{k}$ is given. Under what condition does there exist a nonzero vector $x$ for which $\mathrm{k}(x)=0$ ?

What necessary and sufficient condition must the principal minors of a negative definite quadratic form satisfy?

How many Jordan matrices differing in the multiplicities of the characteristic numbers, in the number and in the sizes of the blocks are there among the matrices of

Find the Jordan form of the matrix and the change-of-basis matrix to the Jordan basis for the transformation given in the initial basis by the matrix