# Question on linear transformaiton

• December 2nd 2011, 01:45 AM
noricka
Question on linear transformaiton
Let V and W be vector spaces over F field, dimV=n, and dim W=m. What is the dimension of the vector space L(V,W), and what is an optional basis?

Thank you!
• December 2nd 2011, 02:05 AM
FernandoRevilla
Re: Question on linear transformaiton
Quote:

Originally Posted by noricka
Let V and W be vector spaces over F field, dimV=n, and dim W=m. What is the dimension of the vector space L(V,W), and what is an optional basis?

Hint : If we fix $B_V$ and $B_W$ basis of $V$ and $W$ respectively, there is a natural isomorphism $\phi:\mathcal{L}(V,W)\to \mathbb{K}^{m \times n}$ given by $\phi (T)=[T]_{B_V}^{B_W}$ .
• December 2nd 2011, 11:28 AM
Deveno
Re: Question on linear transformaiton
you may also wish to consider $E_{i,j} \in \mathcal{L}(V,W) : E_{i,j}(a_1v_1+a_2v_2+\dots+a_nv_n) = a_jw_i$.