Question

9:461 Jerry Math 35100 Homework 2 14 Let A- 31li0 0 Without mtiling the ms (a) Find the dimension of eath of the four lundamental sspaces icd Find a basis for N(4) and for N(A) a) Fnd a basis for 5 What is the dimension of

Answer 1

A = [1 3 0, 0 1 - 1, 0 0 1][1 0 0, 2 0 0, 0 3 0, 3 1 0] A_1 A_2 Observe that det (A_1) = 1 notequalto 0 so A_1 is invertible Thus dim (col (A)) = dim (col (A_2)) dim (Row (A)) = dim (Row (A_2)) dim (N (A)) = dim (N (A_2)) dim (R (A)) = dim (R (A_2)) R (A) = solution space of A. Now dim (cos (A_2)) = dim (Row (A_2)) = 2 (use rank nullity theorem) & dim (N (A_2)) = 1 (only one zero row) so all four fundamental space dim is determined. Observe then Ax = [1 3 0, 0 1 - 1, 0 0 1][x_1 + 2x_2 + 3x_4 3x_2 + x_4 0] = [x_1 + 2x_2 + 3x_4 3x_1 + 6x_2 + 3x_3 + 10x_4 - 3x_3 - x_4] = [b_1 b_2 b
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