Question regarding basis vectors of root reference frame...

Probably my question is rather silly but then again I would rather ask you than going ahead and doing something even sillier.

Right, in an old maths book(or at least what remains of it) I was reading through I found this question and it got me rather confused:

I was wondering if you could help me deduce what was meant by "basis vectors of the root reference frame ([1, 0]^T, [0, 1]^T)"? Did they mean that [1,0]^T is 'y' and [0,1]^T is 'x' on the coordinate system, or did they simply ask us to describe the transformation from one point to the other using the rotation matrix given earlier???

Also regarding the active and passive transformation part of the question, how can I describe the interpretation then?

I know what active and passive transformations are with active transformation, being a transformation which actually changes the physical position of a point which can be defined even in the absence of a coordinate system and that passive is is merely a change in the coordinate system in which the object is described (change of coordinate map, or change of basis). Would that require me to draw another diagram explaining how it happened or what should I do then?

Thank you guys in advance, I really appreciate the help you have given me.

I hope that this figure can help.

As you noticed the matrix $R_\alpha$ represents a rotation of angle $\alpha$.

Applying this matrix to the vectors of the standard basis $\mathbf{i}$ and $\mathbf{j}$ ( called root reference frame in your book) we find a rotated (orthonormal) basis whose basis vectors $\mathbf{i'}$ and $\mathbf{j'}$ have as components , in the standard basis, the columns of $R_\alpha$

So, by linearity, applying the same matrix to a vector $\mathbf{x}$ we obtain the components in the standard basis of the rotated vector $\mathbf{x'}$. And this is an active transformation.

The components of $\mathbf{x}$ in the rotated basis $\{\mathbf{i'},\mathbf{j'}\}$ are the comumns of the inverse matrix $R^{-1}_\alpha$, that is a rotation of angle $-\alpha$, and this is a passive transformation that gives the components of a vector in the new basis.