| course |
course_year |
question_number |
tags |
title |
year |
Linear Mathematics |
IB |
3 |
IB |
2001 |
Linear Mathematics |
|
2.II.15C |
2001 |
Define the dual $V^{}$ of a vector space $V$. Given a basis $\left{v_{1}, \ldots, v_{n}\right}$ of $V$ define its dual and show it is a basis of $V^{}$. For a linear transformation $\alpha: V \rightarrow W$ define the dual $\alpha^{}: W^{} \rightarrow V^{*}$.
Explain (with proof) how the matrix representing $\alpha: V \rightarrow W$ with respect to given bases of $V$ and $W$ relates to the matrix representing $\alpha^{}: W^{} \rightarrow V^{}$ with respect to the corresponding dual bases of $V^{}$ and $W^{*}$.
Prove that $\alpha$ and $\alpha^{*}$ have the same rank.
Suppose that $\alpha$ is an invertible endomorphism. Prove that $\left(\alpha^{}\right)^{-1}=\left(\alpha^{-1}\right)^{}$.