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Complete parts (a) and (b) for the matrix below. A = k= -7 -9 6 1 35 (a) Find k such that Nul(A) is a subspace of Rk.

For the matrix A below, find a nonzero vector in Nul A and a nonzero vector in Col A. A = 124 0 013-2 A nonzero column vector in Nul A is

Complete parts (a) and (b) for the matrix below. 3 7 2 -3 A = a. Find k such that Nul(A) is a subspace of Rk. k =

It can be shown that a solution of the system below is x₁ = 5, x₂ = 4, and x3 = -1. Use this fact and the theory of null spaces and column spaces of matrices to explain why another solution is x₁ = 50, x₂ = 40, (Observe how the solutions are related, but make no other calculations.) 7x1 - 10x1 - 13x1 - 15x2 + 20x2 + 20x₂ - 25x3 = 0 +30x3 = 0 + 15x3 = 0 Let A be the coefficient matrix of the given homogeneous system of equations. The vector x = Next, determine the relationship between the given solution x = 50 40 - 10 (Simplify your answer.) 5 A -1 C 5 4 --- is in the vector space Nul A. 4 and the proposed solution.

Let A = - 10 20 - 5 10 and w= Determine if w is in Col(A). Is w in Nul(A)? Determine if w is in Col(A). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. OA. The vector w is in Col(A) because Ax = w is a consistent system. One solution is x = OB. The vector w is not in Col(A) because Ax = w is an inconsistent system. One row of the reduced echelon form of the augmented matrix [A 0] has the form [0 0 b] where b = OC. The vector w is not in Col(A) because w is a linear combination of the columns of A. O D. The vector w is in Col(A) because the columns of A span R².