Math Problems Vault

Problem 1055

Which of the following maps of the space $\mathbb {R}_{n}[x]$ into itself are linear:

?

(1)

$P(x) \mapsto P(1) \cdot 1$ ;

(2)

$P(x) \mapsto P^{\prime }(x)$ ;

(3)

$P(x) \mapsto \int_{0}^{1} P(t) d t \cdot x^{2}, n \geqslant 2$ ;

(4)

$P(x) \mapsto P(0) P(1) \cdot x, n \geqslant 1$ ;

(5)

$P(x) \mapsto \frac{d^{n}}{d x^{n}} P\left(x^{2}\right)$ ;

(6)

$P(x) \mapsto x^{n} P(1 / x)$ ;

(7)

$P(x) \mapsto \left(\int_{0}^{x} P(t) d t\right) / x$ ?

Problem 1056

Consider the linear space of all convergent numerical sequences $\left\{ a_{n}\right\}$ . Which of the functions $\left\{ a_{n}\right\} \mapsto \lim_{n \rightarrow \infty } a_{n}$ , $\left\{ a_{n}\right\} \mapsto \sup a_{n}$ and $\left\{ a_{n}\right\} \mapsto a_{1}$ are linear?

?

Problem 1057

In the linear space $\mathrm{Mat}_{n}$ of real matrices of order $n$ , consider the trace and determinant functions. Are they linear?

?

Problem 1058

Prove that for any nonzero linear function $l$ on a finite-dimensional linear space $V$ there exists a basis in which the function $l$ has the form $l(x) = x^{1}$ , where $x^{1}$ is the first coordinate of the vector $x \in V$ in this basis.

?

Problem 1059

Prove that the map inverse to a linear isomorphism is also a linear isomorphism. Deduce from this that the relation of being isomorphic is an equivalence relation.

?

Problem 1060

Prove that the linear functions on an arbitrary vector space $V$ also form a linear space (which is denoted by $V^{*}$ and is called the conjugate or dual space to the space $V$ ). Prove that for any finite-dimensional space $V$ there is a natural isomorphism between the spaces $V$ and $\left(V^{*}\right)^{*}$ .

?

Problem 1061

Let $V$ be a linear space with basis $e_{1}, \ldots , e_{n}$ . Prove the existence and uniqueness of linear functions $e^{1}, \ldots , e^{n}$ on $V$ such that $e^{i}\left(e_{j}\right) = \delta_{j}^{i}$ (the Kronecker symbol). Verify that such a set forms a basis of the space $V^{*}$ of linear functions on the space $V$ (such a basis is called the dual or conjugate basis to the basis $e_{1}, \ldots , e_{n}$ ). How is it related to the coordinates in the basis $e_{1}, \ldots , e_{n}$ ?

?

Problem 1062

Let $e_{1}, \ldots , e_{n}$ be a basis of the space $V$ . Prove that every linear function $l$ on $V$ has the form $l(x) = l_{1} x^{1}+\ldots +l_{n} x^{n}$ , where $x^{1}, \ldots , x^{n}$ are the coordinates of the vector $x$ in the basis $e_{1}, \ldots , e_{n}$ . What coordinates does the function $l$ have in the dual basis $e^{1}, \ldots , e^{n}$ of the space $V^{*}$ ?

?

Problem 1063

Let $l$ be a nonzero linear function on a linear space $V$ (not necessarily finite-dimensional). Prove that the kernel $\operatorname {Ker}(l)$ of the function $l$ , i.e., the set of all vectors $x \in V$ for which $l(x) = 0$ , is a maximal nontrivial linear subspace of $V$ , i.e., it is not contained in any subspace of the space $V$ other than itself and $V$ .

Prove that the space $V / \operatorname {Ker}(l)$ is one-dimensional.

?

Problem 1064

Prove that $k$ linear functions on an $n$ -dimensional linear space are linearly independent if and only if the intersection of their kernels is an $(n-k)$ -dimensional subspace.

?

Problem 1065

Prove that if the kernels of two linear functions $l_{1}$ and $l_{2}$ on a linear space $V$ (not necessarily finite-dimensional) coincide, then there exists a nonzero number $\lambda$ such that $l_{1}(x) = \lambda l_{2}(x)$ for every $x \in V$ .

?

Problem 1066

Prove that to every linear map А: $V \rightarrow W$ there naturally corresponds a linear map $\mathrm{A}^{*}: W^{*} \rightarrow V^{*}$ .

Let $e_{1}, \ldots , e_{m}$ and $f_{1}, \ldots , f_{n}$ be bases of the spaces $V$ and $W$ respectively, and let $e^{1}, \ldots , e^{m}$ and $f^{1}, \ldots , f^{n}$ be the dual bases to them in $V^{*}$ and $W^{*}$ . How are the matrices $A$ and $A^{*}$ of the linear maps $\mathbf{A}$ and $\mathbf{A}^{*}$ respectively related in these bases?

?

Problem 1067

The annihilator $\operatorname {Ann} W$ of a subset $W$ of a vector space $V$ is the set of all linear functions vanishing on $W$ . Show that $\operatorname {Ann} W$ is a linear subspace of the space $V^{*}$ of linear functions, and that if $V$ is finite-dimensional, then $\operatorname {dim}(\operatorname {Ann} W) = \operatorname {dim} V-\operatorname {rk} W$ .

?

Problem 1068

Prove that, if $W_{1}$ and $W_{2}$ are subspaces of some finite-dimensional vector space, then the equality $\operatorname {Ann} W_{1} = \mathrm{Ann} W_{2}$ is equivalent to the coincidence of the subspaces $W_{1}$ and $W_{2}$ . Verify the relations $\operatorname {Ann}\left(W_{1}+W_{2}\right) = \operatorname {Ann} W_{1} \cap \operatorname {Ann} W_{2}$ and $\operatorname {Ann}\left(W_{1} \cap W_{2}\right) = \operatorname {Ann} W_{1}+\operatorname {Ann} W_{2}$ .

?

Problem 1069

Let $W$ be a linear subspace of some finite-dimensional vector space. Prove that $\operatorname {Ann}(\operatorname {Ann}(W)) = W$ .

?

Problem 1070

Let $W$ be a subspace of a vector space $V$ complementary to a subspace $U$ . Prove that $W$ is isomorphic to the quotient space $V / U$ .

?

Problem 1071

Prove that for any linear subspace $U$ of a space $V$ there is an isomorphism between the spaces $\operatorname {Ann}(U)$ and $(V / U)^{*}$ .

?

Problem 1072

Show that the set of all semilinear functions on a finite-dimensional space $V$ over the field $\mathbb {C}$ has the structure of a linear space over $\mathbb {C}$ of dimension $n = \operatorname {dim} V$ . Show also that a function $l$ is semilinear if and only if the function $\bar{l}$ , given by the formula $\bar{l}(x) = \overline{l(x)}$ , is linear.

?

Problem 1073

Let $e_{1}, \ldots , e_{n}$ be a basis of a complex linear space $V$ , and let $e^{1}, \ldots , e^{n}$ be the dual basis. Show that the functions $\overline{e^{1}}, \ldots , \overline{e^{n}}$ form a basis of the space of semilinear functions. Let $C$ be the transition matrix to a new basis $f_{1}, \ldots , f_{n}$ in the space $V$ . Find the transition matrix from the basis $\overline{e^{1}}, \ldots , \overline{e^{n}}$ to the new dual basis $\overline{f^{1}}, \ldots , \overline{f^{n}}$ .

?

Problem 1074

In the space $\mathbb {R}_{n}[t]$ of polynomials of degree at most $n$ , consider the linear functions $l^{i}, i = 0,1, \ldots , n$ , given by the formula

l^{i}(P) = \int _{0}^{i+1} P(t) d t

Prove that these functions form a basis of the space of linear functions on $\mathbb {R}_{n}[t]$ .

?

Problem 1075

In the space $\mathbb {R}_{n}[t]$ of polynomials of degree at most $n$ , consider the linear functions $l^{i}, i = 0,1, \ldots , n$ , given by the formula $l^{i}(P) = P^{(i)}\left(t_{0}\right)$ . Prove that these functions form a basis of the space of linear functions on $\mathbb {R}_{n}[t]$ . Find the dual basis $P_{0}, \ldots , P_{n}$ in the space $\mathbb {R}_{n}[t]$ .

?

Problem 1076

In the space $\mathbb {R}_{n}[t]$ of polynomials of degree at most $n$ , consider the linear functions $l^{i}, i = 0,1, \ldots , n$ , given by the formula $l^{i}(P) = P\left(t_{i}\right)$ , where $P(t) \in \mathbb {R}_{n}[t]$ , and $t_{0}, t_{1}, \ldots , t_{n}$ are distinct points of the number line. Prove that these functions form a basis of the space of linear functions on $\mathbb {R}_{n}[t]$ . Find the dual basis in the space $\mathbb {R}_{n}[t]$ .

?

Problem 1077

Prove that the vector space $(\mathbb {R}^{t})^{*}$ of linear functions on the space $\mathbb {R}^{t}$ of all polynomials is not isomorphic to the space $\mathbb {R}^{t}$ .

?

Problem 1078

In the space $\mathrm{Mat}_{n}(\mathbb {C})$ of complex square matrices of order $n$ , consider the linear functions $l^{p q}, p, q = 1, \ldots , n$ , given by the formula $l^{p q}(A) = \operatorname {tr}\left(U^{p} V^{q} A\right)$ , where $A \in M_{n}$ ,

U =\left(\begin{smallmatrix} e^{2 \pi i / n} & 0 & \ldots & 0 \\ 0 & e^{4 \pi i / n} & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \ldots & 0 & 1 \end{smallmatrix}\right), \quad V =\left(\begin{smallmatrix} 0 & 0 & \cdots & 0 & 1 \\ 1 & 0 & \cdots & 0 & 0 \\ 0 & 1 & \cdots & 0 & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & \cdots & 0 & 1 & 0 \end{smallmatrix}\right)

Prove that these functions form a basis of the space of linear functions on $\mathrm{Mat}_{n}(\mathbb {C})$ . Find the dual basis $(A_{p q})$ in the space $\mathrm{Mat}_{n}(\mathbb {C})$ .

?

Problem 1079

The linear map $\mathrm{A}: \mathbb {R}^{3} \rightarrow \mathbb {R}^{2}$ is given by the matrix

\left(\begin{smallmatrix} 2 & 3 & -1 \\ 1 & -2 & 1 \end{smallmatrix}\right)

In the domain, a new basis $e_{1} = (1,-1,0)$ , $e_{2} = (0,2,-1), e_{3} = (1,0,1)$ is chosen. Find the matrix of the map А in this basis.

?

Problem 1080

Find the matrix of the map from the previous problem, if in the codomain a new basis is chosen: $f_{1} = (3,2), f_{2} = (1,1)$ , and the basis in $\mathbb {R}^{3}$ is left unchanged.

?