Linear transformation between vector spaces over $\mathbb{Z}_5$

Question

Let $f:\mathbb{Z}_{5}^{3} \rightarrow \mathbb{Z}_{7}^{2}$ be a transformation such that $$f\left(\begin{bmatrix}x\\y\\z\end{bmatrix}\right):=\begin{bmatrix}x+2z\\x+y-z\end{bmatrix}$$

My homework first asks me to show this is a linear transformation and decide whether or not it's surjective. This $f$ transformation is then used in several questions which suggest its linearity (finding its Kernel, finding its transformation matrix, etc.).

My confusion is that the question doesn't specify what field these vector spaces are on. So the question would be if, say, I decide to pick $\mathbb{Z}_{5}$ as the field I work with, is there any way to make sense of this? after all, the operations in my two vector spaces are defined differently (namely, using their respective mod equivalence classes).

If it helps, the context of this homework is diagonalization and triangularization of transformation matrices.

Answer 1

I think there is probably a typo, as I don't see how to talk about a linear transformation when the codomain is a vector space over a different field than the domain.

Let's say $f:\Bbb Z_5^3\to\Bbb Z_5^2$.

The matrix is $\begin{pmatrix}1&0&2\\1&1&-1\end{pmatrix}$.

Row-reduce: $\to\begin {pmatrix}1&0&2\\0&1&2\end {pmatrix}$. It has rank $2$.

Thus it's surjective.

The kernel is $\{\begin{pmatrix}-2t\\-2t\\t\end{pmatrix}\,,t\in\Bbb Z_5\}$.

Answer 2

Starting point:

So let $u,v\in \mathbb{Z}^3_5$ (where $u=(u_1,u_2,u_3)^T$) then you need to show $$ f((u+v)\mod5) = (f(u) + f(v)) \mod 7 $$ as clearly $f(u), f(v) \in \mathbb{Z}^2_7$, where each element of the vectors $u$ and $v$ are mapped according to your function $f$ defined above (where the right hand side summation should be interpreted as $\mod 7$); and for $\alpha \in \mathbb{Z}$ then $$ f(\alpha u \mod 5) = \alpha f(u) \mod 7 $$