What do you mean by scaling a vector in a particular direction?

Question

Find a 2 × 2 matrix $A$ such that the linear transformation $T : R^2 → R^2$ defined by $T(~u) =A~u$ geometrically scales by a factor of 2 in the direction of the vector $\left(1,2\right)^T$ and scales by the factor of -3 in the direction of $\left(-1,1\right)^T$.

Above is the question I'm trying to solve. I don't understand what the question means when it says "scales by a factor in a particular direction."

Any help in understanding this would really help!

Answer 1

It means $T(a(1,\,2)^T+b(-1,\,1)^T)=2a(1,\,2)^T-3b(-1,\,1)^T$.

Answer 2

It says that $(1,2)^t$ is an eigenvector with eigenvalue $2$ and $(-1,1)^t$ is an eigenvector with eigenvalue $-3$. Since you have the eigenvectors and eigenvalues you can readily express $A$ in its diagonalized form.