Linear functions and mappings

Let $V_{0} \xrightarrow {\mathcal{A}_{1}} V_{1} \xrightarrow {\mathcal{A}_{2}} \ldots \xrightarrow {\mathcal{A}_{m}} V_{m}$ be a sequence of linear maps of vector spaces. Prove that

Let $F$ be a field of $q$ elements. Find:

Let a linear map $\mathcal{A}: V \rightarrow W$ have, in the bases $(e_{1}, e_{2}, e_{3})$ of the space $V$ and $\left(f_{1}, f_{2}\right)$ of the space $W$, the matrix $\left(\begin{array}{lll}0 & 1 & 2 \\ 3 & 4 & 5\end{array}\right)$. Find the matrix of the map $\mathcal{A}$ in the bases $(e_{1}, e_{1}+e_{2}, e_{1}+e_{2}+e_{3})$ and $\left(f_{1}, f_{1}+f_{2}\right)$.

Let $L=K[x]_{1}$ (see 34.12), $K$ a field. Find the matrix of the linear map $\mathcal{A}: f(x) \mapsto f(S)$ of the space $L$ into the space $M=\mathbf{M}_{2}(K)$, where $S=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$ is a fixed matrix, if the basis chosen in $L$ is $(1, x)$, and the basis in $M$ is the basis of matrix units.

Let $\mathcal{A}, \mathcal{B}: V \rightarrow W$ be linear maps with $\operatorname {dim}(\operatorname {Im} \mathcal{A}) \leqslant \operatorname {dim}(\operatorname {Im} \mathcal{B})$. Prove that there exist operators $\mathcal{C}, \mathcal{D}$ on $V$ and $W$ such that $\mathcal{A}=\mathcal{D B C}$, and $\mathcal{C}$ (or $\mathcal{D}$) may be chosen nonsingular.

Let $\mathcal{A}, \mathcal{B}: V \rightarrow W$ be linear maps. Prove that the following conditions are equivalent:

Let $\mathcal{A}, \mathcal{B}: V \rightarrow W$ be linear maps. Prove that the following conditions are equivalent:

Let $\mathcal{A}: V \rightarrow W$ be a linear map. Prove that there exists a linear map $\mathcal{B}: W \rightarrow V$ such that $\mathcal{A}=\mathcal{A B} \mathcal{A}$, $\mathcal{B}=\mathcal{B} \mathcal{A B}$.

Let $V=\mathbb {R}\left[x\right]_{n}$, and let the maps $\alpha^{a}(a \in \mathbb {R}), \beta^{i}, \gamma^{i}$ of the space $V$ into $\mathbb {R}$ be given by the rules

Prove that the systems:

Prove that for any nonzero linear function $f$ on an $n$-dimensional space $V$ there exists a basis $(e_{1}, \ldots , e_{n})$ of $V$ such that

for any coefficients $x_{1}, \ldots , x_{n}$.

Prove that every $k$-dimensional subspace of an $n$-dimensional space is the intersection of the kernels of some $n-k$ linear functions.

Let $f$ be a nonzero linear function on a vector space $V$ (not necessarily finite-dimensional), $U=\operatorname {Ker} f$. Prove that:

Prove that if two linear functions on a vector space have the same kernel, then they differ by a scalar factor.

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

Prove that the vectors $e_{1}, \ldots , e_{k}$ of a finite-dimensional space $V$ are linearly independent if and only if there exist linear functions $f^{1}, \ldots , f^{k} \in V^{*}$ such that $\operatorname {det}\left(f^{i}\left(e_{j}\right)\right) \neq 0$.

For every subset $U$ of a finite-dimensional space $V$ and every subset $W$ of the dual space $V^{*}$, set

Prove that:

Prove that the space of polynomials $\mathbb {Q}\left[x\right]$ is not isomorphic to its dual.

Let $l_{1}, l_{2}$ be two linear functions on a linear space $V$ such that $l_{1}(x) l_{2}(x)=0$ for all $x \in V$. Prove that one of the functions is zero.

Let $l_{1}, \ldots , l_{k}$ be linear functions on a linear vector space $V$ over an infinite field. Prove that if $l_{1}(x) \ldots l_{k}(x)=0$ for all $x \in V$, then one of the functions is zero.

Let $K$ be a finite field of $q^{n}$ elements, $F$ a subfield of $K$ of $q$ elements, $l(x)=x+x^{q}+\ldots +x^{q^{n-1}}, x \in K$. Prove that: