Linear composition

Question

can you help me with this quest?

About composition $f$ and vector space $\mathbf{V}=\mathbb{Z^4_2}$ we know the following:

$f \circ f = id_V$,$~~f $ $ \left(\begin{array}{ccc} 1\\ 0\\ 1\\ 0\\ \end{array}\right)= \left(\begin{array}{ccc} 1\\ 1\\ 0\\ 0\\ \end{array}\right) $, $~~f $ $ \left(\begin{array}{ccc} 0\\ 1\\ 0\\ 1\\ \end{array}\right)= \left(\begin{array}{ccc} 1\\ 1\\ 0\\ 1\\ \end{array}\right) $

$id_V$ is identity.

Find $f~((x_1,x_2,x_3,x_4)^T).$

I would be gratefull for any kind of advice.

Thanks

Answer 1

You are given values of $f$ on two vectors; the fact that $f\circ f = id_V$ should allow you to get values of $f$ on two additional vectors. Now you know what $f$ does to four vectors in a four-dimensional vector space; that is (typically) enough information to specify $f$ completely. In particular, if you can write a vector $\vec x$ as a linear combination of those four vectors, then you should be able to evaluate $f(\vec x)$.

Answer 2

Hint: The matrix of the transformation with respect to the basis $(1,0,1,0)^T,(1,1,0,0)^T,(0,1,0,1)^T,(1,1,0,1)^T$ is $$ \pmatrix{ 0&1&0&0\\ 1&0&0&0\\ 0&0&0&1\\ 0&0&1&0 } $$