Spaces and Span * Spaces and Span 1. (1 point) Add section header Recall that for a set of vectors vi. v2.... . in the vector space V, if W is the set of all linear combinations of the vectors vi. V2. ... V.. W is a subspace of V and W is spanned by v1. Vy . . . V.- Showing a vector, w. is in the span ofvi. v. ...v. requires showing w is a linear combination of vi, vy. ... .. One way to do this is to allow for a matrix A = [ v1. v. ... .] and augment it with the vector in question B = [A. w] = [VI. v2. ... V, w] then compare the rank of the original matrix with the augmented one. Consider the following vectors: * If we are given a vector space spanned by S=(v_1, v_2} v_1 = [1; -2; 2]; * (3x1) vector v_2 = [2; 2; 1]; * (3x1) vector * We want to determine if w_1 and w_2 are in the span of {v_1, v_2}. w_1 = [6; 0; 6]; % (3x1) vector w_2 = [6; 0; 5]; % (3x1) vector A = [v_1, v_2]; rank_A = rank(A); * Since v_1 and v_2 are linearly independent, the rank of A is 2. * Now, we can compare the rank of A with the augmented matrix rank for w_1 * then w_2 to see if they are in the span of {v_1, v_2). rank_wl = rank([A, w_1]) % rank_wl = 2 rank_w2 = rank([A, w_2]) % rank_wl = 3 So, since rank (A) = 2 and the rank with w_1 was also 2, w_1 is in the span of S = { v1, w2). However, the rank of the augmented matrix with w_2 was 3, so w_2 is not in the span. 2. (10 points) Complete the following: . Create the vectors , = -2 and ex - and use them to create a matrix A = [v_1, v_2]. . Determine if wi = and w1 = ONN are in the span of S = {vi.} using rank()- Run your code and make a comment about if w or w, are in the span of S = {vi. } or not and how you know