Dual space

As easily follows from the definition (see Exercise 1 from § 1), the trace function $\operatorname {tr}: X \mapsto \operatorname {tr} X$ is linear on the space $V=M_{n}(\mathfrak {K})$ of all square matrices of order $n$ over the field $\mathfrak {K}$. Prove that every linear function $f$ on $V$ has the form $f(X)=\operatorname {tr} A X$, where the matrix $A=A_{f}$ is uniquely determined.

Let $a(t)$ be a fixed polynomial from $\mathbb {R}{}[t], P_{n}$ the subspace of real polynomials of degree $\leqslant n-1$. Consider the following functions on $P_{n}$:

Let $V$ be a vector space, and let functions $f, g \in V^{*}$ be such that $\operatorname {Ker} f=\operatorname {Ker} g$. Prove that then $g=\lambda f$ for some scalar $\lambda$.

Let $\mathbf{x}$ be a nonzero vector of the space $V$. Is the function $f \in V^{*}$ uniquely determined by the condition $f(\mathbf{x})=1$?

Prove that for every nonzero linear function $f$ on an $n$-dimensional vector space $V$ over $\mathfrak {K}$ there exists a basis $\left(\mathbf{e}_{1}, \ldots , \mathbf{e}_{n}\right)$ of the space $V$ such that