Sorry about the goofy thread title, it should read "Linear Functionals and Direct Sums".
Let be a finite-dimensional vector space over the field of scalars , and a linear map from to . Show that
Well, I reasoned that if T is not the zero map, then the range of T is (simple to show), in which case the range of T is spanned by one vector. From dim(V)=dim(kerT)+dim(ranT) we must have that the dimension of the kernel is one less than that of V. So let be a basis for kerT. Extending it to a basis to V we only need add , which is not in the kernel, because if it was it would contradict the linear independence of the base for the kernel.
So a basis for V is . Obviously .
The fact that is a basis for shows that . Thus, .
If is the zero map then the set of vectors not in the kerel is the empty set, and so , and everything is all good.
Is this right?