Question: 1) Let V ={(a+b, b, b+c, b) : a R, b R, c R }. It is a sub-space of R 4 . a) Find a basis for V. (No need to show work.). (b) what is the

Question:

1) Let V ={(a+b, b, b+c, b) : a ∈ R, b ∈ R, c ∈ R }. It is a sub-space of R4.

a) Find a basis for V. (No need to show work.).        (b)  what is the dimension of V?       (c)  what is the dimension of the orthogonal complement of V? (i.e., the set of all ⃗v ∈ R4 such that ⃗v is orthogonal to V).               (d) Find an orthogonal basis for V . (Show all work).

2) Let (1) and (2) be two systems of linear equations.

                         X + 2Y + 3Z = 8

  (equation 1)   X - 2Y +4Z = 15             ( equation 2)   X - 3Y = 1

                         X + 3Y + Z = 1                                        4Y - Z = 1

Find the solutions of (1) and (2) and describe them as sets of points in R3.

3) Given two vectors u = [1  -1  2]T and v = [3  -1  2]T compute the following.

(a) Projection of u on v.              (b) u × v.      (c) Projection of u on u × v .        (d) Area of the parallelogram determined by u and v.  

                                                                   (All vectors have direction →)

4) Let P = (2, 1, 3) a point and let L be the line through P with direction vector →u = [2, −2, 1]T . Let Λ be the plane given with equation X + 2Y - Z = 7.

(a) Find the intersection of the line L and the plane Λ or prove that they have no points in common.    (b) Find the distance between P and Λ and the point on Λ closest to P .