Hey I have a problem with this exercise. Can someone help me?
The erxercise is to show that there is a $\mathbb{Q}$-linear map $f : \mathbb{Q}^2 → \mathbb{Q}^2$ with $f( \begin{pmatrix} 1\\ 0 \end{pmatrix} ) = \begin{pmatrix} 1\\ 4 \end{pmatrix}$ and $f( \begin{pmatrix} 1\\ 1 \end{pmatrix}) = \begin{pmatrix} 2\\ 5 \end{pmatrix}$. Determine $f ( \begin{pmatrix} 2\\ 3 \end{pmatrix})$.Is $f$ bijective ?
So for the first I said that since the vectors $\begin{pmatrix}1\\0\end{pmatrix}$, $\begin{pmatrix}1\\4\end{pmatrix}$ and $\begin{pmatrix}1\\1\end{pmatrix}$, $\begin{pmatrix}2\\5\end{pmatrix}$ are linearly independent there exists a linear map. To find $f(\begin{pmatrix}2\\3\end{pmatrix})$ is quite easy.
The problem is that I don't know how to prove that $f$ is bijective.
Can someone help me?