Linear Functionals and Direct Sums.
Hi everyone
Sorry about the goofy thread title, it should read "Linear Functionals and Direct Sums".
Let $\displaystyle V$ be a finite-dimensional vector space over the field of scalars $\displaystyle F$, and $\displaystyle T$ a linear map from $\displaystyle V$ to $\displaystyle F$. Show that
$\displaystyle V = kerT\oplus span\{ u\}$
where $\displaystyle Tu \neq 0$
Well, I reasoned that if T is not the zero map, then the range of T is $\displaystyle F$ (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 $\displaystyle \{u_1,\ldots ,u_k \}$ be a basis for kerT. Extending it to a basis to V we only need add $\displaystyle u$, 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 $\displaystyle \{u_1,\ldots ,u_k, u \}$. Obviously $\displaystyle kerT \cap span\{ u\} = 0$.
The fact that $\displaystyle \{u_1,\ldots ,u_k, u \}$ is a basis for $\displaystyle V$ shows that $\displaystyle V = kerT + span\{ u\}$. Thus, $\displaystyle V = kerT\oplus span\{ u\}$.
If $\displaystyle T$ is the zero map then the set of vectors not in the kerel is the empty set, and so $\displaystyle span\{ u\}=\{ 0\}$, and everything is all good.
Is this right?
