Question: 1) Let V be the set of all vectors [a b c d] T in R 4 satisfying a+c = b+d. a) Is V a vector subspace of R 4 (circle the correct answer)?

Question:

1)  Let V be the set of all vectors [a  b  c  d]T in R4 satisfying a+c = b+d.

a) Is V a vector subspace of R4 (circle the correct answer)?                     (YES   NO)

1. b) If your answer to part (a) is "no" provide a proof. It your answer is "yes," explain why and find the dimension and a basis of V.

2)  Find an orthogonal basis of R3 that contains the vector (2, 1, −2). Show all work.

3) Find the distance between the point P = (2, 0, 5) and the plane 3x − 6y + 2z = 6. Find the point on the plane closest to P.

4) Let ⃗v = (1, 2, −2) and define T : R3 → R3 by T(⃗u) = proj⃗v (⃗u). Find matrix A such that T (⃗u) = A⃗u. Show all work.