Let T: R^2 → R^2 be defined by T(x, y) = (2x - y, x + 3y). Find the standard matrix of T.

Step 1: Recall how to find the standard matrix of a linear transformation.

For a transformation T: R^2 → R^2, the columns of the standard matrix are T(e1) and T(e2), where:

e1 = (1, 0)
e2 = (0, 1)

Step 2: Find T(e1).

Substitute x = 1 and y = 0:

T(1, 0) = (2(1) - 0, 1 + 3(0))

T(1, 0) = (2, 1)

Step 3: Find T(e2).

Substitute x = 0 and y = 1:

T(0, 1) = (2(0) - 1, 0 + 3(1))

T(0, 1) = (-1, 3)

Step 4: Use these vectors as the columns of the matrix.

The first column is (2, 1), and the second column is (-1, 3).

Therefore, the standard matrix is:

[[2, -1], [1, 3]]

Step 5: Verify the matrix representation.

Multiplying the matrix by (x, y) gives:

[[2, -1], [1, 3]](x, y) = (2x - y, x + 3y)

This matches the definition of T.

Therefore, the standard matrix of T is [[2, -1], [1, 3]].