Find the matrix of the following transformations:
(a) Multiplication of the point (x, y, z) in ℝ^3 to give the point (x - y, y + 2z, x + z)
(b) Reflection of points in ℝ^2 across the line in y = x.
To find the matrix representation of a given transformation, we can determine the image of the standard basis vectors and use them as columns to construct the matrix.
(a) Multiplication of the point (x, y, z) in ℝ^3 to give the point (x - y, y + 2z, x + z):
Let's consider the standard basis vectors: e1 = (1, 0, 0), e2 = (0, 1, 0), and e3 = (0, 0, 1).
Applying the given transformation to each of these vectors, we get:
T(e1) = (1 - 0, 0 + 2(0), 1 + 0) = (1, 0, 1)
T(e2) = (0 - 1, 1 + 2(0), 0 + 0) = (-1, 1, 0)
T(e3) = (0 - 0, 0 + 2(1), 0 + 1) = (0, 2, 1)
Now, we can form the matrix with these image vectors as columns:
[1 -1 0]
[0 1 2]
[1 0 1]
Therefore, the matrix representation of the given transformation is:
[1 -1 0]
[0 1 2]
[1 0 1]
(b) Reflection of points in ℝ^2 across the line y = x:
Similarly, let's consider the standard basis vectors in ℝ^2: e1 = (1, 0) and e2 = (0, 1).
Applying the reflection transformation to each of these vectors, we get:
T(e1) = (0, 1)
T(e2) = (1, 0)
Now, we can form the matrix with these image vectors as columns:
[0 1]
[1 0]
Therefore, the matrix representation of the given transformation is:
[0 1]
[1 0]