# Subspace

• January 14th 2007, 09:16 AM
thesummerofgeorge
Subspace
I need help proving the following:

Let F be any field. Let A be any m x n over F. Suppose that W is a subspace of F^n. Show that the set {Aw:w in W) is a subspace of F^m.

Note that the lowercase w in the above set is the w vector.

Thank you for any help.
• January 14th 2007, 09:59 AM
ThePerfectHacker
Quote:

Originally Posted by thesummerofgeorge
I need help proving the following:

Let F be any field. Let A be any m x n over F. Suppose that W is a subspace of F^n. Show that the set {Aw:w in W) is a subspace of F^m.

Note that the lowercase w in the above set is the w vector.

Thank you for any help.

Let $S=\{ A \omega | \omega \in W \}$
Again the idea is the same as I said.
We need to show closure.
$kA \omega =A (k\omega)=A \omega_1 \in S$
Where $\omega_1=k\omega \in W$
(Because $\omega$ is an element of a vector space over F thus, $k\omega \in W$.)

$A \omega_1+A\omega_2=A(\omega_1+\omega_2)=A \omega_3 \in S$
Where, $\omega_3 \in W$.
Thus, the set $S$ which is a subset of $W$ is closed under scalar multiplication und vector addition, that is it must be a vector space (or subspace of W) over the field F.