1. Solve Equations
a.
x1 + 2x2 + 2x3 = 4
x1 + 3x2 + 3x3 = 5
2x1 + 6x2 + 5x3 = 6
b.
2x1 + 4x2 - 3x3 + 5x4 + x5 = 9
3x1 + x2 + x4 - 3x5 = 0
2x1 + 7x2 - 5x3 + 2x4 + 2x5 = -3
Consider the set of 5 vectors, S, from R^4
S = {(1, 1, 3, 1), (2, 1, 2, 1), (7, 3, 5, 5), (1, 1, -1, 1), (-1, 0, 0, 0)}
and consider the infinite set of vectors <S> formed from all possible linear combinations of the elements of S.
a. Is u = (-15, -6, 19, 5) an element of S?
b. Is v = (3, 1, 2, -1) an element of S?
c. Find a base of S.
3. Let x = [2, 1, 3]^T, y = [3, -1, 4]^T, and z = [2, 6, 4]^T
(a) What is the dimension of span(x, y, z)? Justify your answer.
(b) Give a geometric description of span(x, y, z).
4. T = SPAN(x + 2, x + 1, x^2 - 1) in P_3. What is the dimension of T? Justify your solution.
5. Determine the null space of A = [[1, 2, -3, -1], [-2, -4, 6, 3]].
6. What is the definition of "S is a subspace of a vector space V"? Use this definition to show that S = {(x, y, z)^T | z = x + y} is a subspace of R^3.
7. Let A = [[1, 2, 2, 3, 1, 4], [0, 0, 2, 3, 4, 9], [0, 0, 3, 6, 8, 9]]. Find a basis for the column space of A. What is the rank of A?
8. Show that v1 = [1, 2, 3]^T, v2 = [-2, 1, 0]^T, v3 = [1, 0, 1]^T can be a basis for R^3.
9. Let A be an m x n matrix. Show that if A has linearly independent column vectors, then N(A) = {0}.
