For which vector $y$ does the equation $Ax = y$ have a solution $x$?

Question

I have a linear algebra problem given a matrix $A$ which is $5×5$ and the problem asked this question: For which vector $y$ does the equation $Ax = y$ have a solution $x$? ($x$ and $y$ have vector notation) What does this mean? Can somebody let me know? Thank you!

Answer 1

See here:
System of linear equations - Consistency

Your system has a solution if it's consistent.

According to the Rouché–Capelli theorem, any system of linear equations (overdetermined or otherwise) is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix. If, on the other hand, the ranks of these two matrices are equal, the system must have at least one solution. The solution is unique if and only if the rank equals the number of variables.

See also this theorem I think in my country it's known as Kronecker theorem. But now I see that in English it's better to call it Rouché–Capelli theorem.

Answer 2

The product $Ax$ can be viewed as a linear combination of the columns of $A$, with coefficients given by the entries of $x$. (It’s a fairly simple but worthwhile exercise to prove this for yourself.) This means that for any $x$ whatsoever, $Ax$ lies in the column space of $x$. What’s more, since $x$ is arbitrary, we can come up with an $x$ for any arbitrary linear combination of those columns. Therefore, the equation $Ax=y$ can be solved iff $y$ is an element of $A$’s column space. So, to answer this question, you’re probably meant to described this space in some more explicit way.