Impossible. If is a basis of , then
is a generator system of so,
.
Fernando Revilla
This may be a stupid question, but when I was studying the rank-nullity theorem, I was wondering about the next problem:
The rank-nullity theorem states, for a linear map T: V --> W:
"dim(Im T) + dim(Ker T) = dim V"
But what happens if you, for example, have a linear map from R² to R³. Then the dimension of the image is 3 and the dimension of V is 2. But that means that the dimension of the kernel has to be -1, and that's not possible, is it?
I hope someone can help me out with this.
Impossible. If is a basis of , then
is a generator system of so,
.
Fernando Revilla
No, that's impossible. Are you clear on what the "image" of a linear map is? For T:V-> W, the image of T is a subspace of W but not necessarily all of W.
No, it is not possible- and so you have proved that the dimension of the image of T cannot be larger than the dimension of V (but can be less).and the dimension of V is 2. But that means that the dimension of the kernel has to be -1, and that's not possible, is it?
I hope someone can help me out with this.