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Math 51, Spring 2011 Henry Adams, April 18 These are the problems we solved during my review session for midterm 1. (1) Suppose v ∈ Rn is a unit vector (that is, kvk = 1). Prove that for any vector w ∈ Rn , the vector w − (w · v)v is orthogonal to v. (2) Let T : Rn → Rn be a linear transformation and let V = {x ∈ Rn | T (x) = 5x}. Prove that V is a linear subspace of Rn . (3) Suppose v1 , v2 , and v3 are nonzero vectors that are orthogonal to each other. Prove that the set {v1 , v2 , v3} is linearly independent. 3 0 −3 12 0 1 2 3 10 0 (4) Let A = 2 1 0 11 1. 0 1 2 3 1 (a) Find rref(A). (b) Find nullity(A) and rank(A). (c) Find N (A) and a basis for N (A). (d) Find C(A) and a basis for C(A). 1 2 (e) Find all solutions x to Ax = Ac, where c = 3. 4 5 (f) Is there a vector b ∈ R4 such that Ax = b has no solutions x? (g) Is there a vector b ∈ R4 such that Ax = b has exactly one solution x? (5) Consider the following linear subspace of R4 . ( x1 ) x2 x1 −2x3 +3x4 = 0 V = x3 x2 −x3 =0 x4 (a) Find a matrix A such that N (A) = V . (b) Find B such C(B) = V . that a matrix −2 4 (6) Let v = −1 and w = 2 . 1 1 (a) Find the angle between v and w. (b) Find the area of the triangle with vertices v, w, and ~0. (7) Let {u, v, w} be a basis for a subspace V of Rn . Is {u − v, v − w, u − w} a basis for V ? (8) Let T : R2 → R2 be the linear transformation that satisfies −3 0 1/2 1/2 T = 1 and T = 1 . 1/2 −1/2 0 2 Find matrix A such that T (x) = Ax for all x ∈ R2 . (9) Consider the system of equations x − 3y = b1 3x + ay = b2 where b1 and b2 are fixed constants. For which values of a does the system of equations have exactly one solution? 1 1 (10) Find a parametric equation for the plane in R3 that passes through the points 2, 3 2 4 7 5, and 8 . 6 10