solve the given problems 13. Let A = 1 0 3 1 2 -1 1 -2 (a)

solve the given problems

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13. Let A = 1 0 3 1 2 -1 1 -2 (a) Taking X = = (1, 1, -2) shows that (-1,0,4)= AX is in the range of A. Show that (-1,0, 4) is a linear combination of (1,3,-1), (0, 1, 1), and (1,2,-2). (b) The vector v = = 2(1,3,-1) + (0, 1, 1)-3(1,2,-2) = (-1,1,5) is a linear combi- nation of the columns of A. Find X so that AX = v. (c) Is the vector (3,7,-5) in the subspace spanned by (1,3,-1), (0, 1, 1), and (1,2,-2)? (d) Show that (1, 2, -2) is a linear combination of (1,3,-1) and (0,1,1). (e) Show that R(A) is the subspace spanned by (1,3,-1) and (0, 1, 1). 14. Let E= 2 0 -1 1 1 1 1 3 0 1 1 2 1 -2 2 3 -2 1 1 2 (a) Find vectors that span N(E) the nullspace of E. (b) Find vectors that span R(E) the range of E. (c) Find vectors that span N(E) the nullspace of E'. (d) Find vectors that span R(E') the range of E'. 13. Let A = 1 0 3 1 2 -1 1 -2 (a) Taking X = = (1, 1, -2) shows that (-1,0,4)= AX is in the range of A. Show that (-1,0, 4) is a linear combination of (1,3,-1), (0, 1, 1), and (1,2,-2). (b) The vector v = = 2(1,3,-1) + (0, 1, 1)-3(1,2,-2) = (-1,1,5) is a linear combi- nation of the columns of A. Find X so that AX = v. (c) Is the vector (3,7,-5) in the subspace spanned by (1,3,-1), (0, 1, 1), and (1,2,-2)? (d) Show that (1, 2, -2) is a linear combination of (1,3,-1) and (0,1,1). (e) Show that R(A) is the subspace spanned by (1,3,-1) and (0, 1, 1). 14. Let E= 2 0 -1 1 1 1 1 3 0 1 1 2 1 -2 2 3 -2 1 1 2 (a) Find vectors that span N(E) the nullspace of E. (b) Find vectors that span R(E) the range of E. (c) Find vectors that span N(E) the nullspace of E'. (d) Find vectors that span R(E') the range of E'.

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