For each of the following linear transformations, write down its matrix and describe the transformation

a) g(x,y)=(4x,6y)
b) h(x,y)=(x+2y,y)
c) k(x,y)=(y,x)
so I have worked out the matrices:
(4 0
0 6)

(1 2
0 1)

(0 1
1 0)
Not sure what the transformations would be?

To describe the transformations represented by the given matrices, let's consider the effect of each transformation on the standard basis vectors in 2-dimensional space:

a) Matrix: (4 0
0 6)

The first column of this matrix represents the transformation of the vector (1, 0), and the second column represents the transformation of the vector (0, 1).

- The first column (4, 0) indicates that the x-coordinate is multiplied by 4, while the y-coordinate remains unaffected.
- The second column (0, 6) indicates that the y-coordinate is multiplied by 6, while the x-coordinate remains unaffected.

Therefore, the transformation can be described as stretching the x-coordinate by a factor of 4, and the y-coordinate by a factor of 6. This results in a stretched representation of the original shape.

b) Matrix: (1 2
0 1)

Similarly, the first column (1, 0) indicates that the x-coordinate remains unaffected, and the y-coordinate is multiplied by 2.
- The second column (0, 1) indicates that both the x and y-coordinates remain unaffected.

Therefore, this transformation can be described as a shear in the y-direction, where the y-coordinate is scaled by a factor of 2, while the x-coordinate remains unchanged.

c) Matrix: (0 1
1 0)

The first column (0, 1) indicates that the x-coordinate is now equal to the original y-coordinate, and the y-coordinate becomes the original x-coordinate.
- The second column (1, 0) indicates that the x-coordinate becomes the original y-coordinate, and the y-coordinate is now equal to the original x-coordinate.

Therefore, this transformation swaps the x and y-coordinates. It flips the shape along the line y = x.