Question 1:
1.) Let V = span((1, 1, 0), (0, 0, 1)). Let prv : R3 → R3 denote the linear transformation whose value on a vector w∈R3 is equal to prv (w), the orthogonal projection of w onto V.
(a) Use the Dimension Theorem (Theorem 12.1 from the note) to prove that the nullity of prv is equal to 1.
(b) Find the standard matrix A of prv : R3 → R3. Show your work.
Attached Theorem 12.1 from the note:
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- Attachment 2
- Attachment 3
Notice that if F has standard matrix A, then F is invertible if and only if A isinvertible and the standard matrix of F-1 is A-1. Moreover the composition oftwo invertible linear transformations is also an invertible linear transformation.Fo G(x, y, z, w) = ABENGIThe set of invertible linear transformations from R" to R" is known as thegeneeral linear group GL(n, R) and has important applications in physics andgeometry. Recall from Theorem 8.3 that an n x n matrix A is invertible ifNul A = {0} or Col A = R". Applying this to the standard matrix of a lineartransformation F: R" -> R" and using Theorem 10.2 we see that F is invertibleENG Hif and only ker(F) = {0} or R(F) = R".=(x+ 2y - 2z + w,y - 2, 2 + w).Example 12.3. Let F, G: R2 -> R2 be defined by F(x, y) = (x + 4y, 2x + 8y),G(x, y) = (4x - 8y, -x+ 2y). Then we compareFo G(x, y) = F(4x - 8y, -x+ 2y)= ((4x - 8y) + 4(-x + 2y), 2(4x - 8y) + 8(-x + 2y))= (0,0)withGoF (x, y) = G(x+ 4y, 2x + 8y)= (4(x + 4y) - 8(2x + 8y), -(x + 4y) + 2(2x + 8y))= (-12x - 48y, 3x + 12y).Given two linear transformations F, G: R" -> R", if the standard matricesof F and G are A and A-1 respectively, then the standard matrix of F o G issimplyAA-1 = In ,and soFoG(u) = Inu =u VuER".This leads to the concept of invertible linear transformations.A linear transformation F: R" -> R" is invertible if there is a linear transfor-mation G: R" -, R" such thatFo G(u) = u= GoF(u)Vu ER"In such a case G, the inverse of F, is unique and we denote it by by F-1
