# show that P is a subspace..

• August 4th 2011, 04:18 PM
Oiler
show that P is a subspace..
Hey all, another subspace related question..

how can I show that if $P$ is a subspace of $R^m$, then $U = \{\vec{u} \in R^n: A\vec{u} \in P\}$ is a subspace of R^n, where A is an m*n matrix.

Thanks.
• August 4th 2011, 07:16 PM
FernandoRevilla
Re: show that P is a subspace..
(i) $A0=0\in P$ (because $P$ is subspace). Hence $0\in U$.

(ii) For all $u,v\in U$ we have $Au\in P$ and $Av\in P$. Then $A(u+v)=Au+Av\in P$ (because $P$ is subspace) . Hence $u+v\in U$ .

(iii) ... Try it.