1. Consider Let R(A) column space of A R(AT) column space of AT, = = A = (b) Find all the vectors b = 12 1 0 0 251 1 0 372 2 -2 1 4 4 9 3 (a) By finding the linearly independent columns, express the four fundamental spaces: R(A), R(AT), N(A), and N(AT). Also find their dimensions. b₁ b₂ N(A) = null space of A, N(AT) = null space of AT. ER such that the system Ax = b is consistent. b3 b4 [Try to establish a relation in the components of b.] Using this result, decide for -2 3 what value of k € R the system Ax = b is inconsistent where b [Instruction: Do not solve Ax = b explicitly.] (c) Do you see any relation between the vector b for which Ax N(AT)? Can you prove this relation in general? -1 k = b is consistent and

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