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I need help for these linear algebra problems

3:11 PM Tue 21 Jun 2 0 2 77% O X 88 Q n Introduction to Linear Algebra + X . .. - 3.4. Independence, Basis and Dimension 175 Problem Set 3.4 Questions 1-10 are about linear independence and linear dependence. Show that v1, v2, v3 are independent but v1, v2, v3, 4 are dependent: Solve Civ1 + C202 + C303 + CAVA = 0 or Ax = 0. The u's go in the columns of A. 2 (Recommended) Find the largest possible number of independent vectors among 3 Prove that if a = 0 ord = 0 or f = 0 (3 cases), the columns of U are dependent: 4 If a, d, f in Question 3 are all nonzero, show that the only solution to Ux = 0 is = 0. Then the upper triangular O has independent columns. 5 Decide the dependence or independence of (a) the vectors (1, 3, 2) and (2, 1, 3) and (3, 2, 1) (b) the vectors (1, -3, 2) and (2, 1, -3) and (-3, 2, 1). 6 Choose three independent columns of U. Then make two other choices. Do the same for A 2 3 4 1 2 3 4 0 6 7 0 U= 0 0 09 and A = 0 6 7 0 0 0 0 9 4 6 8 2 7 If w1, w2, w3 are independent vectors, show that the differences v1 = w2 - w3 and U2 = W1 - w3 and v3 = W1 - w2 are dependent. Find a combination of the u's that gives zero. Which matrix A in [ v1 v2 v3 ] = [w1 w2 w3 ] A is singular? 8 If w1, w2, w3 are independent vectors, show that the sums v1 = w2 + w3 and v2 = wj + w3 and v3 = w1 + w2 are independent. (Write c1v1 + C202 + C3v3 = 0 in terms of the w's. Find and solve equations for the c's, to show they are zero.)