Math Problems Vault

Problem 39.1

Which of the following maps in the corresponding vector spaces are linear operators:

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(a)

$x \mapsto a$ ( $a$ is a fixed vector);

(b)

$x \mapsto x+a$ ( $a$ is a fixed vector);

(c)

$x \mapsto \alpha x$ ( $\alpha$ is a fixed scalar);

(d)

$x \mapsto (x, a) b$ ( $V$ is a Euclidean space, $a, b$ are fixed vectors);

(e)

$x \mapsto (a, x) x$ ( $V$ is a Euclidean space, $a$ is a fixed vector);

(f)

$f(x) \mapsto f(a x+b) \quad \left(f \in \mathbb {R}\left[x\right]_{n} ; a, b\right.$ are fixed numbers);

(g)

$f(x) \mapsto f(x+1)-f(x)\left(f \in \mathbb {R}\left[x\right]_{n}\right)$ ;

(h)

$f(x) \mapsto f^{(k)}(x) \quad \left(f \in \mathbb {R}\left[x\right]_{n}\right)$ ;

(i)

$\left(x_{1}, x_{2}, x_{3}\right) \mapsto \left(x_{1}+2, x_{2}+5, x_{3}\right)$ ;

(j)

$\left(x_{1}, x_{2}, x_{3}\right) \mapsto \left(x_{1}+3 x_{3}, x_{2}^{3}, x_{1}+x_{3}\right)$ ;

(k)

$\left(x_{1}, x_{2}, x_{3}\right) \mapsto \left(x_{1}, x_{2}, x_{1}+x_{2}+x_{3}\right)$ ?

Problem 39.2

Prove that every linear operator maps any linearly dependent system of vectors to a linearly dependent system.

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Problem 39.3

Prove that in an $n$ -dimensional space, for any linearly independent system of vectors $a_{1}, \ldots , a_{n}$ and an arbitrary system of vectors $b_{1}, \ldots , b_{n}$ , there exists a unique linear operator taking $a_{i}$ to $b_{i}(i=1, \ldots , n)$ .

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Problem 39.4

Prove that in a one-dimensional vector space every linear operator has the form $x \mapsto \alpha x$ , where $\alpha$ is some scalar.

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Problem 39.5

Find the images and kernels of the linear operators from problem 39.1 (for the parameter values at which the corresponding map is a linear operator):

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(a)

$x \mapsto a$ ( $a$ is a fixed vector, $a=0$ );

(b)

$x \mapsto x+a$ ( $a$ is a fixed vector, $a=0$ );

(c)

$x \mapsto \alpha x$ ( $\alpha$ is a fixed scalar);

(d)

$x \mapsto (x, a) b$ ( $V$ is a Euclidean space, $a, b$ are fixed vectors);

(e)

$x \mapsto (a, x) x$ ( $V$ is a Euclidean space, $a$ is a fixed vector, $a=0$ );

(f)

$f(x) \mapsto f(a x+b) \quad \left(f \in \mathbb {R}\left[x\right]_{n} ; a, b\right.$ are fixed numbers);

(g)

$f(x) \mapsto f(x+1)-f(x)\left(f \in \mathbb {R}\left[x\right]_{n}\right)$ ;

(h)

$f(x) \mapsto f^{(k)}(x) \quad \left(f \in \mathbb {R}\left[x\right]_{n}\right)$ ;

(i)

$\left(x_{1}, x_{2}, x_{3}\right) \mapsto \left(x_{1}, x_{2}, x_{1}+x_{2}+x_{3}\right)$ .

Problem 39.6

(a)(b)

Prove that the differentiation operator:

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(a)

is singular on the space of polynomials of degree $\leqslant n$ ;

(b)

is nonsingular on the space of functions with basis $(\cos t, \sin t)$ .

Problem 39.7

(a)(b)

Prove that every subspace of a vector space is:

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(a)

the kernel of some linear operator;

(b)

the image of some linear operator.

Problem 39.8

Prove that if two linear operators of rank 1 have equal kernels and equal images, then they commute.

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Problem 39.9

Let $\mathcal{A}-F$ -linear operator on a subspace $L$ of a space $V$ , with $L$ different from $V$ . Prove that there are infinitely many linear operators on $V$ whose restriction to the subspace $L$ coincides with $\mathcal{A}$ , provided the field $F$ is infinite.

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Problem 39.10

(a)(b)

Let $\mathcal{A}$ be a linear operator on a space $V, L$ a subspace of $V$ . Prove that:

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(a)

the image $\mathcal{A}(L)$ and the full preimage $\mathcal{A}^{-1}(L)$ are subspaces of $V$ ;

(b)

if the operator $\mathcal{A}$ is nonsingular and $V$ is finite-dimensional, then

\operatorname {dim} \mathcal{A}(L)=\operatorname {dim} \mathcal{A}^{-1}(L)=\operatorname {dim} L

Problem 39.11

Let $\mathcal{A}$ be a linear operator on a space $V, L$ a subspace of $V$ , and $L \cap \operatorname {Ker} \mathcal{A}=0$ . Prove that any linearly independent system of vectors from $L$ is taken by the operator $\mathcal{A}$ to a linearly independent system.

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Problem 39.12

Prove Frobenius's inequality for linear operators $\mathcal{A}, \mathcal{B}, \mathcal{C}$ :

\operatorname {rk}(\mathcal{B} \mathcal{A})+\operatorname {rk}(\mathcal{A} \mathcal{C}) \leqslant \operatorname {rk} \mathcal{A}+\operatorname {rk}(\mathcal{B} \mathcal{A} \mathcal{C})

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Problem 39.13

A linear operator $\mathcal{A}$ is called a pseudoreflection if $\operatorname {rk}(\mathcal{A}-\mathcal{E})=1$ .

Prove that in an $n$ -dimensional space every linear operator is a product of at most $n$ pseudoreflections.

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Problem 39.14

Prove that for a linear operator $\mathcal{A}$ on an $n$ -dimensional space the set of operators $\mathcal{X}$ such that $\mathcal{A X}=0$ is a vector space, and find its dimension.

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Problem 39.15

Find the matrix of the operator:

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(a)

$\left(x_{1}, x_{2}, x_{3}\right) \mapsto \left(x_{1}, x_{1}+2 x_{2}, x_{2}+3 x_{3}\right)$ on the space $\mathbb {R}^{3}$ in the basis of unit vectors;

(b)

of the rotation of the plane through an angle $\alpha$ in an arbitrary orthonormal basis;

(c)

of the rotation of three-dimensional space through an angle $2 \pi / 3$ about the line given in a rectangular coordinate system by the equations $x_{1}=x_{2}=x_{3}$ , in the basis of unit vectors of the coordinate axes;

(d)

of the projection of three-dimensional space onto the coordinate axis of the vector $e_{2}$ parallel to the coordinate plane of the vectors $e_{1}$ and $e_{3}$ , in the basis $(e_{1}, e_{2}, e_{3})$ ;

(e)

$x \mapsto (x, a) a$ in a Euclidean space, in the orthonormal basis $(e_{1}, e_{2}, e_{3})$ , where $a=e_{1}-2 e_{3}$ in the indicated basis;

(f)

$X \mapsto \left(\begin{array}{ll}a & b \\ c & d\end{array}\right) X$ on the space $\mathbf{M}_{2}(\mathbb {R})$ in the basis of matrix units;

(g)

$X \mapsto X\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$ on the space $\mathbf{M}_{2}(\mathbb {R})$ in the basis of matrix units;

(h)

$X \mapsto { }^{t} X$ on the space $\mathbf{M}_{2}(\mathbb {R})$ in the basis of matrix units;

(i)

$X \mapsto A X B$ ( $A, B$ are fixed matrices) on the space $\mathrm{M}_{2}(\mathbb {R})$ in the basis consisting of matrix units;

(j)

$X \mapsto A X+X B(A, B-$ fixed matrices) on the space $\mathbf{M}_{2}(\mathbb {R})$ in the basis of matrix units;

(k)

of differentiation on the space $\mathbb {R}\left[x\right]_{n}$ in the basis $\left(1, x, \ldots , x^{n}\right)$ ;

(l)

of differentiation on the space $\mathbb {R}\left[x\right]_{n}$ in the basis $\left(x^{n}, x^{n-1}, \ldots , 1\right)$ ;

(m)

of differentiation on the space $\mathbb {R}\left[x\right]_{n}$ in the basis

\left(1, x-1, \frac{(x-1)^{2}}{2}, \ldots , \frac{(x-1)^{n}}{n!}\right)

Problem 39.16

Prove that in the space $\mathbb {R}^{3}$ there exists a unique linear operator taking the vectors $(1,1,1),(0,1,0)$ , $(1,0,2)$ respectively to the vectors $(1,1,1),(0,1,0),(1,0,1)$ , and find the matrix of this operator in the basis consisting of unit vectors.

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Problem 39.17

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 $\left(a_{1}, \ldots , a_{n}\right)$ .

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Problem 39.18

Find the general form of the matrices of linear operators on an $n$ -dimensional space taking given linearly independent vectors $a_{1}, \ldots , a_{k}(k<n)$ to given vectors $b_{1}, \ldots , b_{k}$ , in a basis of the form $\left(a_{1}, \ldots , a_{k}, a_{k+1}, \ldots , a_{n}\right)$ .

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Problem 39.19

(a)(b)

Let a linear operator on a space $V$ have, in the basis $\left(e_{1}, \ldots , e_{4}\right)$ , the matrix

\left[\begin{smallmatrix} 0 & 1 & 2 & 3 \\ 5 & 4 & 0 & -1 \\ 3 & 2 & 0 & 3 \\ 6 & 1 & -1 & 7 \end{smallmatrix}\right]

Find the matrix of this operator in the bases:

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(a)

$\left(e_{2}, e_{1}, e_{3}, e_{4}\right)$ ;

(b)

$\left(e_{1}, e_{1}+e_{2}, e_{1}+e_{2}+e_{3}, e_{1}+e_{2}+e_{3}+e_{4}\right)$ .

Problem 39.20

Let a linear operator on the space $\mathbb {R}\left[x\right]_{2}$ have, in the basis $\left(1, x, x^{2}\right)$ , the matrix

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

Find its matrix in the basis

\left(3 x^{2}+2 x+1, x^{2}+3 x+2,2 x^{2}+x+3\right)

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Problem 39.21

Let a linear operator on the space $\mathbb {R}^{3}$ have, in the basis

((8,-6,7),(-16,7,-13),(9,-3,7))

the matrix

\left[\begin{smallmatrix} 1 & -18 & 15 \\ -1 & -22 & 20 \\ 1 & -25 & 22 \end{smallmatrix}\right]

Find its matrix in the basis

((1,-2,1),(3,-1,2),(2,1,2))

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Problem 39.22

Let a linear operator $\mathcal{A}$ on an $n$ -dimensional vector space $V$ take the linearly independent vectors $a_{1}, \ldots , a_{n}$ to the vectors $b_{1}, \ldots , b_{n}$ respectively. Prove that the matrix of this operator in some basis $\mathbf{e}=\left(e_{1}, \ldots , e_{n}\right)$ equals $B A^{-1}$ , where the columns of the matrices $A$ and $B$ consist respectively of the coordinates of the given vectors in the basis e.

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Problem 39.23

(a)(b)

Find the general form of the matrix of a linear operator $\mathcal{A}$ in a basis whose first $k$ vectors form:

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(a)

a basis of the kernel of the operator $\mathcal{A}$ ;

(b)

a basis of the image of the operator $\mathcal{A}$ .

Problem 39.24

Prove that if $f(t)=f_{1}(t) f_{2}(t)$ is a factorization of the polynomial $f(t)$ into coprime factors and the linear operator $\mathcal{A}$ satisfies $f(\mathcal{A})=0$ , then in some basis the matrix of the operator $\mathcal{A}$ has the form $\left(\begin{array}{cc}A_{1} & 0 \\ 0 & A_{2}\end{array}\right)$ , where $f_{1}\left(A_{1}\right)=0, f_{2}\left(A_{2}\right)=0$ .

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