I would like to ask how to find all linear transformations from $\mathbb Z_2^2$ to $\mathbb Z_2^2$ and how to determine which ones are bijections. Thanks for reply :) .
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Please use mathjax. – Dietrich Burde Dec 28 '16 at 21:09
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For the second part see this duplicate. – Dietrich Burde Dec 28 '16 at 21:15
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After a basis of $\mathbf F_2^2$ has been chosen, endomorphisms of $\mathbf F_2^2$ correspond to invertible matrices in $\mathcal M_2(\mathbf F_2)$, i.e. to matrices
$$\begin{pmatrix}a&b\\c&d \end{pmatrix},\quad a,b,c,d\in \mathbf F_2. $$
Automorphisms correspond to matrices with determinant $1$, or with non-zero, distinct, column vectors since the only elements in $\mathbf F_2$ are $0$ and $1$. Hence the list of such matrices is
\begin{align}&\begin{pmatrix}1&1\\0&1\end{pmatrix},&&\begin{pmatrix}1&0\\0&1 \end{pmatrix},&&\begin{pmatrix}1&1\\1&0\end{pmatrix},&&\begin{pmatrix}1&0\\1&1 \end{pmatrix},
&&\begin{pmatrix}0&1\\1&0\end{pmatrix},&&\begin{pmatrix}0&1\\1&1 \end{pmatrix}.
\end{align}
Bernard
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When you say "linear transformations", I'm going to assume that you mean maps that preserve the module structure of $\left(\mathbb{Z}_2\right)^2$ over $\mathbb Z$. Since a linear map is uniquely determined by its action on a basis (when one exists, of course), we can find all the endomorphisms of $\left(\mathbb{Z}_2\right)^2$ by enumerating all the possible ways we can map $B = \{(1,0),(0,1)\}$ into $\left(\mathbb{Z}_2\right)^2$. The maps which are bijections are the ones for which $f(B)$ is a basis for $\left(\mathbb{Z}_2\right)^2$. I'll leave it up to the reader to create a table.
silvascientist
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