2011-39.md

course	course_year	question_number	tags	title	year
Linear Algebra	IB	39	IB 2011 Linear Algebra	Paper 1, Section II, G	2011

Let $V, W$ be finite-dimensional vector spaces over a field $F$ and $f: V \rightarrow W$ a linear map.

(i) Show that $f$ is injective if and only if the image of every linearly independent subset of $V$ is linearly independent in $W$.

(ii) Define the dual space $V^{}$ of $V$ and the dual map $f^{}: W^{} \rightarrow V^{}$.

(iii) Show that $f$ is surjective if and only if the image under $f^{}$ of every linearly independent subset of $W^{}$ is linearly independent in $V^{*}$.