T R2 rightarrow R2 is a vertical shear transformation that

T : R^2 rightarrow R^2 is a vertical shear transformation that maps e_1 into e_1 - 2e_2 but leaves the vector e_2 unchanged.

Solution

The e1 and e2 are the vectors of size 2X1 i.e. [1,0] and [0,1]

Since the final answer is a 2X1 matrix, hence the dimension of T will be 2X2 matrix

Let the 2X2 T matrix be

T(e1) = e1 - 2e2 = [1,0] - 2 * [0,1] = [1,-2]

multiplying T with e1 we get

a(1) + b(0) = 1 => (a=1)

c(1) + d(0) = -2 => (c=-2)

T(e2) = e2 = [1,0]

multiplying T with e1 we get

a(0) + b(1) = 1 => (b=1)

c(0) + d(1) = 0 => (d=0)

Hence the final T matrix will be equal to

a	b
c	d