Question

LINEAR ALGEBRA- prove this theorem: For nite-dimensional vector spaces V and W and a linear transformation
L : V ! W, we have,
dim(ker(L)) + dim(Im(L)) = dim(V )

Answer 1

that for finite* and L: V-->W

Answer 2

Use the fact that ker L and Im L are linear subspaces of V and W, then choose appropriate bases.

Answer 3

true ya im looking through my notes on that just really confusing

Answer 4

In fact you have \[\mathrm{Im}\;L \simeq V/(\mathrm{ker}\;L)\] which gives you the dimension property as a direct corollary.

Answer 5

ya true dat thx