I was wondering how do you do these two questions?
I can handle this one
First, a definition a shear, in case you need it. A shear parallel to a given line through the origin is a transformation in which the component of a vector perpendicular to the line is unchanged, and the component parallel to the line is increased by an amount proportional to the perpendicular component. If the proportionality constant is k, we call it a k–shear.Okay, so what you are looking at here is a linear transformation that applies both a shear, and a rotation, on the cartesian plane. Recall that we can represent any linear transformation with a matrix multiplication, we just need to find a matrix A such that:
How do we do this? Fortunately, we can actually get each column of the matrix A by considering the transformation of the standard basis vectors in the domain, \(\binom{1}{0}, \binom{0}{1}\). The results in the co-domain will give us the columns of matrix A.
I'll do the first one slow. First, apply the shear to \(\binom{1}{0}\):
Now, apply the rotation:
Let me know if you need help with those individual parts of this question! So, the two vectors we need are:
So the matrix is formed with these as its columns:
There is a fair bit of decently complex theory at play here, let me know if you need anything clarified!
I'll leave you to tackle the second one, same principle, it is just applying the rotation
first!