Question

Show your work explain your reasoning!-Unsupported intermediate or final answers may result in lost marks. Consider the linear transformation T: R^3 rightarrowᴿ³ given by reflection about the P:x + 2y - z = 0, (T)v = v - 2proj_n v, where n is any norm vector for P and v elementof R^3. (a) Find the standard matrix a = [T] for T.

Answer 1

To find the standard matrix for the linear transformation T, apply T to each of the standard basis vectors in R^3. Calculate the projection of each vector onto the normal vector n, and then subtract the projections from the original vectors to find the transformed vectors. The resulting vectors will be the columns of the standard matrix a = [T].

Step-by-step explanation:

To find the standard matrix for the linear transformation T, we need to determine how T changes the standard basis vectors in R^3. The standard basis vectors for R^3 are [1, 0, 0], [0, 1, 0], and [0, 0, 1].

Applying the transformation T to each of these basis vectors, we get:

T([1, 0, 0]) = [1, 0, 0] - 2proj_n([1, 0, 0])

T([0, 1, 0]) = [0, 1, 0] - 2proj_n([0, 1, 0])

T([0, 0, 1]) = [0, 0, 1] - 2proj_n([0, 0, 1])

Using the equation for reflection, we can calculate the projection of each vector onto the normal vector n. Once we have the projections, we can subtract them from the original vectors to find the transformed vectors. The resulting vectors will be the columns of the standard matrix a = [T].