Find an Example for a linear map $T: \mathbb{F}^4 \to \mathbb{F}^4$ such that $ImT = KerT = sp\{(1, 1, 1, 1), (1, 1, 1, 0)\}$

Question

Find an Example for a linear map $T: \mathbb{F}^4 \to \mathbb{F}^4$

such that $$ImT = KerT = sp\{(1, 1, 1, 1), (1, 1, 1, 0)\}$$

My Attempt:

First I completed the two vectors $\{(1, 1, 1, 1), (1, 1, 1, 0)\}$ to a base of $\mathbb{F}^4$, so I picked: $\{(1, 0, 0, 0), (0, 0, 1, 0)\}$

So I'm looking for a map T such that $$ T((1, 1, 1, 1) = T((1, 1, 1, 0)) = 0 ; \ T((1,0, 0, 0)) = (1, 1, 1, 1), \ T(( 0 , 0 , 1, 0)) = (1, 1, 1, 0) $$

Now, Let $(x, y, z, w) \in \mathbb{F}^4$.

This is where I got stuck. what does the vector $(x, y, z, w)$ needs to be existing in order that I could find a linear map $T$ as needed?

Answer 1

Hint

In the basis you chosed (I haven’t verified that it is indeed a basis), the matrix of a linear map satisfying the required condition is

$$T=\begin{pmatrix} 0&0&1&0\\ 0&0&0&1\\ 0&0&0&0\\ 0&0&0&0 \end{pmatrix}$$

Then use a change of basis.

Answer 2

$T(1,0,0,0)$ is the first column vector of $[T]$, while $T(0,0,1,0)$ is the third column vector. Thus, your second and third conditions imply that the first column of $[T]$ is $(1,1,1,1)^T$ and its second column is $(1,1,1,0)^T$.

For $T(1,1,1,1)=T(1,1,1,0)=0$, you require that all rows of $[T]$ should be orthogonal to $(1,1,1,1),(1,1,1,0)$. Therefore,

$$[T]=\begin{bmatrix}1&-2&1&0\\1&-2&1&0\\1&-2&1&0\\1&-1&0&0\end{bmatrix}$$