Linear functions

What conditions single out linear functions from other linear mappings?

How does the row of coefficients of a linear function transform under a change of basis?

How are the coefficients of a linear function in the basis $\mathbf{e}$ expressed in terms of the basis vectors?

Write out the row of coefficients of the zero linear function.

Can the following hold, for a linear function $\mathrm{f}$ given on $\mathcal{L}_{n}$, for all $x\in \mathcal{L}_{n}$:

A linear function $\mathrm{f}$ on $\mathcal{L}_{n}$ and a number $\alpha$ are given. Does there always exist a vector $x$ in $\mathcal{L}_{n}$ such that $\mathrm{f}(x) = \alpha$?

Determine the set of values of an arbitrary linear function on a real vector space.

Let $\left(\xi_{1}, \xi_{2}, \xi_{3}\right)^{T}$ be the coordinate column of a vector $x\in \mathcal{L}_{3}$ in some basis. Will the function $\mathrm{f}$ on $\mathcal{L}_{3}$, defined by the equality, be linear:

Write out the row of coefficients of the function $\mathrm{f}$ in cases 1, 4 of problem 31. 8.

In some basis of the space $\mathcal{L}_{3}$, the functions $\mathrm{f}$ and $\mathrm{g}$ have coordinate rows $(1, 2, 3)$ and $(3, 2, 1)$, respectively. Find the coordinate rows of the functions:

The function $\operatorname {tr} X$ assigns to each square matrix $X$ of order $n$ its trace. Verify that this function is linear, and find its coordinate row (coordinate matrix) in the standard basis of the space of matrices.

Let $C$ be a square matrix of order $n$. Assign to each square matrix $X$ of order $n$ the number $\operatorname {tr}\left(C^{T} X\right)$. Show that this defines a linear function on the space $\mathcal{R}_{n \times n}$, and find its coordinate row (coordinate matrix).

Let f be some linear function defined on the space $\mathcal{R}_{n \times n}$. Prove that there exists a square matrix $C$ such that for an arbitrary matrix $X\in \mathcal{R}_{n \times n}$ the equality $\mathrm{f}(X) = \operatorname {tr}\left(C^{T} X\right)$ holds.

Let a linear function $\mathrm{f}$ on the space $\mathcal{R}_{n \times n}$ satisfy the condition $\mathrm{f}(AB) = \mathrm{f}(BA)$ for any two square matrices $A$ and $B$ of order $n$. Prove that $\mathrm{f}$ is given by the equality $\mathrm{f}(X) = \alpha \operatorname {tr} X$.

Assign to each polynomial $p(t)$ of degree $\leqslant n$ its value at $t = 0$. Prove that this defines a linear function on $\mathcal{P}^{(n)}$, and compute its coordinate row in the basis $1, t, t^{2}, \ldots , t^{n}$.

Let $t_{0}$ be a fixed number. Assign to each polynomial $p(t)$ of degree $\leqslant n$ its value at $t = t_{0}$. Prove that this defines a linear function $\varphi$ on the space $\mathcal{P}^{(n)}$. Compute the coordinate row of the function $\varphi$ in the bases

Let $t_{1}, \ldots , t_{n + 1}$ be pairwise distinct points of the number line, and let $\varphi_{1}, \ldots , \varphi_{n + 1}$ be the linear functions on the space $\mathcal{P}^{(n)}$ corresponding to these points, defined in problem 31. 21.

A linear function $\delta$ assigns to each polynomial $p(t)$ of degree $n\quad (n \leqslant 2)$ its constant term. Decompose this function into a linear combination of the functions $\varphi_{1}, \varphi_{2}, \varphi_{3}$ which assign to each polynomial its value at $t = 1, t = 2$ and $t = 3$, respectively.

Let $t_{0}$ be some, and $t_{1}, \ldots , t_{n + 1}$ pairwise distinct, real numbers. Prove that there exist numbers $\lambda_{1}, \ldots , \lambda_{n + 1}$ such that for any polynomial $p(t) \in \mathcal{P}^{(n)}$ the equality $p\left(t_{0}\right) = \lambda_{1} p\left(t_{1}\right) + \ldots + \lambda_{n + 1} p\left(t_{n + 1}\right)$ holds.

Let $k$ be a natural number. Assign to each polynomial $p(t)$ of degree $\leqslant n$ the value of its $k$-th derivative at $t = 0$. Prove that this defines a linear function on the space $\mathcal{P}^{(n)}$, and compute its coordinate row in the basis $1, t, t^{2}, \ldots , t^{n}$.

Let $k$ be a natural number, $k \leqslant n$, and $t_{0}$ a real number. Assign to each polynomial $p(t)$ of degree at most $n$ the value of its $k$-th derivative at $t = t_{0}$. Prove that this defines a linear function on the space $\mathcal{P}^{(n)}$. Compute its coordinate row in the bases:

Linear functions $\delta_{0}, \delta_{1}, \ldots , \delta_{n}$ are defined on the space $\mathcal{P}^{(n)}$ by the equalities

Prove that the functions $\delta_{0}, \delta_{1}, \ldots , \delta_{n}$ are linearly independent.

The functions $\delta_{0}, \delta_{1}, \ldots , \delta_{n}$ are defined in the same way as in problem 31. 27. Prove that an arbitrary linear function given on the space $\mathcal{P}^{(n)}$ can be decomposed into a linear combination of the functions $\delta_{k}(k = 0, 1, \ldots , n)$.

Let in the basis $e_{1}, e_{2}, e_{3}$ a linear function $f$ be expressed in terms of the coordinates $\xi_{1}, \xi_{2}, \xi_{3}$ of the vector $x$ by the formula $\mathrm{f}(x) = \xi_{1} + 2\xi_{2} + 3\xi_{3}$. By what formula is $\mathrm{f}(x)$ expressed in terms of the coordinates of $x$ in the basis $e_{1}^{\prime } = e_{1} + e_{2}, e_{2}^{\prime } = e_{2} + e_{3}, e_{3}^{\prime } = e_{3} + e_{1}$?

Prove that any nonzero linear function $\mathrm{f}$ on $\mathcal{L}_{n}$ can be reduced, by a suitable choice of basis in $\mathcal{L}_{n}$, to the form $\mathrm{f}(x) = \xi_{1}$, where $\xi_{1}$ is the first coordinate of the vector $x$.

In a basis $e$, a linear function $\mathrm{f}$ has coefficient row $\varkappa$. Find its coefficient row $x^{\prime }$ in the basis $\mathbf{e}^{\prime } = \mathbf{e} S$, if:

The functions $\varphi_{1}, \varphi_{2}, \varphi_{3}$, defined in problem 31. 23, as well as the functions $\delta_{0}, \delta_{1}, \delta_{2}$, defined by means of the formulas

form a pair of bases in the space $\mathcal{P}^{(2) * }$. Write out the transition formulas from the first basis to the second.

How does the biorthogonal basis transform if the given basis is transformed by the transition matrix $S$?

Construct a basis of the space $\mathcal{P}^{(2)}$ biorthogonal to the basis of functions $\varphi_{1}, \varphi_{2}, \varphi_{3}$ defined in problem 31. 23.

Find a basis of the space $\mathcal{P}^{(n)}$ biorthogonal to the basis of functions $\varphi_{1}, \varphi_{2}, \ldots , \varphi_{n + 1}$ constructed in problems $31. 21, 31. 22$. Compute the coordinates of an arbitrary polynomial in the basis found.

Construct a basis of the space $\mathcal{P}^{(2)}$ biorthogonal to the basis of functions $\delta_{0}, \delta_{1}, \delta_{2}$ defined in problem 31. 32.

Let $e_{1}, \ldots , e_{n}$ be a basis in the space $\mathcal{L}_{n}$, and let $\mathrm{f}_{1}, \ldots , \mathrm{f}_{n}$ be the basis in $\mathcal{L}_{n}^{ * }$ biorthogonal to it. Prove that for all $x\in \mathcal{L}_{n}$ the equality holds

and for all y $\in \mathcal{L}_{n}^{ * }$ the equality

Apply formula (1) to the bases considered in problem 31. 34.

Using the result of problem 31. 34, prove that polynomials $p_{0}, \ldots , p_{k}$ of degree at most $k$ are linearly independent if and only if for some $t_{0}$ we have $\operatorname {det}\left\| p_{2}^{(\jmath)}\left(t_{0}\right)\right\| \neq 0$.

Find the basis biorthogonal to the standard basis of the space $\mathcal{R}_{n \times n}$. Compute the matrices $C$ corresponding to the functions of this basis (in the sense of problem 31. 17).

The Pauli matrices

form a basis in the space of complex square matrices of order 2. Find the basis biorthogonal to the basis $\sigma_{0}, \sigma_{1}$, $\sigma_{2}, \sigma_{3}$, and compute the matrices $C$ corresponding to the functions of this basis in the sense of problem 31. 17.

Prove that the product of two linear functions on $\mathcal{L}_{n}$ is identically equal to 0 if and only if at least one of the functions is zero.

Let $\mathrm{f}$ be a linear function on $\mathcal{L}_{n}$. Prove that the set $\mathcal{N}$ of vectors for which $\mathrm{f}(x) = 0$ is a linear subspace of $\mathcal{L}_{n}$. What is the dimension of $\mathcal{N}$? Is it possible for $\mathcal{N}$ to coincide with $\mathcal{L}_{n}$?

Let f, $\mathrm{g}$ be linear functions on $\mathcal{L}_{n}$ and let $\mathrm{f}(x) = 0$ for all those $x$ for which $\mathrm{g}(x) = 0$. Prove that then there exists a number $\alpha$ such that $\mathrm{f} = \alpha \mathrm{g}$.

In the space $\mathcal{L}_{4}$ a basis is chosen, and linear functions with coordinate rows $(5, 24, - 7, - 1)$ and $(- 1, - 2, 7, 3)$ are given. Find the set of vectors on which these functions simultaneously vanish.

Let $\mathcal{N}$ be a linear subspace of $\mathcal{L}_{n}$, and let $\mathcal{K}$ be the set of all linear functions vanishing on $\mathcal{N}$. Prove that $\mathcal{K}$ is a linear subspace of $\mathcal{L}_{n}^{ * }$, and compute its dimension.

A subspace $\mathcal{N}$ in $\mathcal{L}_{5}$ is given in some basis as the linear span of the vectors with coordinate columns $\left[\begin{smallmatrix} 0\\ 0\\ 1\\ 1\\ 1\end{smallmatrix}\right]$ and $\left[\begin{smallmatrix} 0\\ 1\\ 0\\ 0\\ 1\end{smallmatrix}\right]$. Find, in the same basis, the coordinate rows of all linear functions vanishing on $\mathcal{N}$.

A subspace $\mathcal{N} \subset \mathcal{P}^{(6)}$ is given as the set of all polynomials of the form $(t - 1)(t - 2)^{2} p(t)$, where $p(t) \in \mathcal{P}^{(3)}$. Find the set of linear functions defined on $\mathcal{P}^{(6)}$ that vanish on $\mathcal{N}$.

Let $\mathrm{f}_{1}, \ldots , \mathrm{f}_{k}$ and $\mathrm{f}$ be linear functions on a vector space $\mathcal{L}$, and let $\mathcal{N}$ be the set of vectors from $\mathcal{L}$ such that $\mathrm{f}_{1}(x) = \ldots = \mathrm{f}_{k}(x) = 0$. Prove that $\mathrm{f}$ decomposes in terms of $\mathrm{f}_{1}, \ldots , \mathrm{f}_{k}$ if and only if $\mathrm{f}(x) = 0$ for all $x$ from $\mathcal{N}$.