What is the dimension of the kernel in this case?

[Guest]

What is the dimension of the kernel of a linear transformation from infinite dimensional vector space to finite dimensional vector space?

May someone help me? I think the answer is infinite but i don't know how to prove this.

[HallsofIvy]

Since the "image" space is finite dimensional, the subspace of vectors in the "domain" space that are mapped to non-zero vectors must be finite dimensional. That means that all other vectors in this "domain" space are mapped to the zero vector- are in the kernel which must be infinite dimensional.