Let T:V→W be a linear transformation where V is an infinite-dimensional vector space and W is a finite-dimensional vector space. What is the dimension of the kernel?
Hello everyone. Sorry my English, it's not my first language.
I tried doing this:
T(v1)=w1
T(v2)=w2
...
T(vn)=wn
But V has infinite vectors, so remaining vectors are in the kernel.
I don't know if I can use dim V = dim KerT + dim ImT here.
I'm not sure of the things I said. If someone may help me I would appreciate a lot. Thank you very much for attention!
Suppose the dimension of the kernel is finite, so $\ker f$ has $\{y_1,\dots,y_n\}$ as basis.
If $\{f(x_1),\dots,f(x_m)\}$ is a basis of the image of $f$, prove that $$ \{x_1,\dots,x_m,y_1,\dots,y_n\} $$ is a spanning set for $V$ (actually a basis).