Consider the linear transformation T: R^3->R^3 which acts by rotation around the y-axis by an angle of pi, followed by a shear in the x-direction by a factor of 2.
a) Find the matrix for T. Explain your method.
b) What is T(1,2,3)
c) Without calculation, explain whether the matrix you found in a) is invertible. What is the transformation corresponding to the inverse?
a) To find the matrix for T, we can perform the two transformations separately and then multiply their respective matrices.
First, rotation around the y-axis by an angle of π can be represented by the following matrix:
[cos(π) 0 -sin(π)]
[ 0 1 0 ]
[sin(π) 0 cos(π)]
Since cos(π) = -1 and sin(π) = 0, this matrix simplifies to:
[-1 0 0 ]
[ 0 1 0 ]
[ 0 0 -1]
Next, the shear in the x-direction by a factor of 2 can be represented by the following matrix:
[1 2 0]
[0 1 0]
[0 0 1]
Multiplying these two matrices gives us the matrix for T:
[-1 2 0]
[ 0 1 0]
[ 0 0 -1]
b) To find T(1,2,3), we multiply the matrix for T by the vector (1,2,3):
[-1 2 0] [1] [-1*1 + 2*2 + 0*3] [3]
[ 0 1 0] [2] = [0*1 + 1*2 + 0*3] [2]
[ 0 0 -1] [3] [0*1 + 0*2 - 1*3] [-3]
So, T(1,2,3) = (3,2,-3)
c) To determine if the matrix found in part a) is invertible, we can check its determinant. If the determinant is non-zero, then the matrix is invertible.
The determinant of the matrix is -1*(-1)*1 = -1. Since the determinant is not zero, the matrix is invertible.
The inverse transformation corresponds to the inverse of the matrix, which is formed by taking the reciprocal of each element in the diagonal:
[-1 -2 0]
[ 0 1 0]
[ 0 0 -1]
Therefore, the inverse transformation is a rotation around the y-axis by an angle of -π, followed by a shear in the x-direction by a factor of -2.