Give an example of the matrix $f_{G,B}$ of an $f∈L(X,Y)$ that is a surjection?

Question

Let $X$ be a real linear space with $\dim X$= 5 and $Y$ a real linear space with $\dim Y$= 4

Give an example of the matrix $f_{G,B}$ of an $f∈L(X,Y)$ that is a surjection.

I don't understand this question. What I thought of was that I need come up with a basis G and a basis B and let X = something and Y = something else so that I can find a matrix by finding $(X)_{G}$ and $(Y)_{B}$. From what I understand surjective means that Y must map a least one element in X, so I think I need to find a matrix that has a turning point in each row using the information given. Might be very wrong though...

Sorry for rambling, I would really appreciate your help! Thank you so much

Answer 1

You are on the right track I think. You want to think of some map that goes from a $5$ dimensional space onto a $4$ dimensional space.

To help, try thinking of a version of the problem that is easier to visualize. What if you want to map from a $2$ dimensional space subjectively onto a $1$ dimensional space, such as $\mathbb{R}^2 \to \mathbb{R}$? You can think about this geometrically and see that one candidate transformation would just be a kind of "projection" onto the real line. That is, a transformation that sends

$$ \begin{pmatrix} a \\ b\end{pmatrix} \mapsto a $$

This is certainly surjective because all elements of $\mathbb{R}$ are hit. You can do something similar in the situation given in the question. Namely, take a transformation which sends

$$ \begin{pmatrix} a \\ b \\c \\d \\ e\end{pmatrix} \mapsto \begin{pmatrix} a \\ b \\ c \\ d\end{pmatrix} $$

Again, this is clearly surjective. Can you think of a matrix that will accomplish this task? Spoiler below.

You can use \begin{pmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 1 & 0\end{pmatrix}