2017-39.md

course	course_year	question_number	tags	title	year
Linear Algebra	IB	39	IB 2017 Linear Algebra	Paper 1, Section II, F	2017

Let $U$ and $V$ be finite-dimensional real vector spaces, and let $\alpha: U \rightarrow V$ be a surjective linear map. Which of the following are always true and which can be false? Give proofs or counterexamples as appropriate.

(i) There is a linear map $\beta: V \rightarrow U$ such that $\beta \alpha$ is the identity map on $U$.

(ii) There is a linear map $\beta: V \rightarrow U$ such that $\alpha \beta$ is the identity map on $V$.

(iii) There is a subspace $W$ of $U$ such that the restriction of $\alpha$ to $W$ is an isomorphism from $W$ to $V$.

(iv) If $X$ and $Y$ are subspaces of $U$ with $U=X \oplus Y$ then $V=\alpha(X) \oplus \alpha(Y)$.

(v) If $X$ and $Y$ are subspaces of $U$ with $V=\alpha(X) \oplus \alpha(Y)$ then $U=X \oplus Y$.