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QUESTION

# Notice that if F has standard matrix A, then F is invertible if and only if A is invertible and the standard matrix of F-1 is A-1. Moreover the...

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 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&quot; to R&quot; 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&quot;. Applying this to the standard matrix of a lineartransformation F: R&quot; -&gt; R&quot; and using Theorem 10.2 we see that F is invertibleENG Hif and only ker(F) = {0} or R(F) = R&quot;.=(x+ 2y - 2z + w,y - 2, 2 + w).Example 12.3. Let F, G: R2 -&gt; 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&quot; -&gt; R&quot;, 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&quot;.This leads to the concept of invertible linear transformations.A linear transformation F: R&quot; -&gt; R&quot; is invertible if there is a linear transfor-mation G: R&quot; -, R&quot; such thatFo G(u) = u= GoF(u)Vu ER&quot;In such a case G, the inverse of F, is unique and we denote it by by F-1