2014-40.md

course	course_year	question_number	tags	title	year
Linear Algebra	IB	40	IB 2014 Linear Algebra	Paper 4, Section II, G	2014

Let $V$ be a real vector space. What is the dual $V^{}$ of $V ?$ If $e_{1}, \ldots, e_{n}$ is a basis for $V$, define the dual basis $e_{1}^{}, \ldots, e_{n}^{}$ for $V^{}$, and show that it is indeed a basis for $V^{*}$.

[No result about dimensions of dual spaces may be assumed.]

For a subspace $U$ of $V$, what is the annihilator of $U$ ? If $V$ is $n$-dimensional, how does the dimension of the annihilator of $U$ relate to the dimension of $U$ ?

Let $\alpha: V \rightarrow W$ be a linear map between finite-dimensional real vector spaces. What is the dual map $\alpha^{}$ ? Explain why the rank of $\alpha^{}$ is equal to the rank of $\alpha$. Prove that the kernel of $\alpha^{}$ is the annihilator of the image of $\alpha$, and also that the image of $\alpha^{}$ is the annihilator of the kernel of $\alpha$.

[Results about the matrices representing a map and its dual may be used without proof, provided they are stated clearly.]

Now let $V$ be the vector space of all real polynomials, and define elements $L_{0}, L_{1}, \ldots$ of $V^{}$ by setting $L_{i}(p)$ to be the coefficient of $X^{i}$ in $p$ (for each $p \in V$ ). Do the $L_{i}$ form a basis for $V^{}$ ?