Find the standard matrix for the linear transformation T.T(x, y, z) = (x + z, x − z, z − x)

The standard matrix for a linear transformation T: R^n -> R^m is a matrix A such that T(x) = Ax for all x in R^n.

In this case, the linear transformation T: R^3 -> R^3 is given by T(x, y, z) = (x + z, x - z, z - x).

We can write this transformation in terms of column vectors as follows:

T(x, y, z) = T([x, y, z]) = [x + z, x - z, z - x]

This can be rewritten as a matrix multiplication:

T([x, y, z]) = A[x, y, z]

where A is the standard matrix for T.

We can find A by observing how T acts on the standard basis vectors in R^3, which are e1 = [1, 0, 0], e2 = [0, 1, 0], and e3 = [0, 0, 1].

T(e1) = T([1, 0, 0]) = [1 + 0, 1 - 0, 0 - 1] = [1, 1, -1] T(e2) = T([0, 1, 0]) = [0 + 0, 0 - 0, 0 - 0] = [0, 0, 0] T(e3) = T([0, 0, 1]) = [0 + 1, 0 - 1, 1 - 0] = [1, -1, 1]

Therefore, the standard matrix A for the linear transformation T is given by the columns T(e1), T(e2), and T(e3):

A = [T(e1), T(e2), T(e3)] = [[1, 1, -1], [0, 0, 0], [1, -1, 1]]