So I just learned that we can describe vector transformations of shapes using base vectors, where the base vector I = $$ \begin{pmatrix} 1 \\ 0 \\ \end{pmatrix} $$ and J=$$ \begin{pmatrix} 0 \\ 1 \\ \end{pmatrix}$$
The transformation of I and J by matrix \begin{pmatrix} 0 & 1 \\ -1 & 0 \\ \end{pmatrix}
is a clockwise rotation of 90 degrees.
I understand how that's happening. Because if I draw I and J on an xy-axis graph, I can clearly see the points move that way Now I'm supposed to express a reflection in the y-axis, using the help of base vectors.
A reflection in y-axis of base vectors I and J should have them go from (a) to (b) as in the diagram below (with J' superimposing J):
and so I believe the transformation vector, using the base vectors, should be this: \begin{pmatrix} -1 & 0 \\ 0 & 1 \\ \end{pmatrix}
But that's not the case! The correct answer is \begin{pmatrix} -1 & 0 \\ 0 & -1 \\ \end{pmatrix}
How is that so? Why the -1 in the lower right corner? Can someone please explain?