Recall that the dimension of a subspace S is the number of elements in a basis for S. Here are some questions about basis and dimension. (a) Let A = [1 2 3 0 2 3], and define S = {u elementof R^3: Au = 0}. Find a basis of S and compute its dimension. (b) Let S = span{(1, 2, 0, -1), (2, -1, 2, 3), (-1, -11, 6, 13), (4, 3, 2, 1)}. Find a subset of the four vectors defining the span that form a basis for S. (c) Consider the set S of simultaneous solutions to the equations x+y+z+w = 0, x-y+w = 0, in R^4. Find a basis for S and report its dimension. (d) Is the following statement true or false: in R^3, let P_1, P_2 be two distinct planes containing 0. Then the dimension of their intersection is 1. (e) A hyperplane H Subset R^4 is, by definition, a subspace of dimension 3. If H_1 and H_2 are hyperplanes in R^4 then so is their intersection H_1 Intersection H_2. What are the possible values of dim(H_1 Intersection H_2)? Justify your answer with examples.
