1. Consider 121 0 0 251 1 0 A= 372 2 -2 493-1 4 Let R(A)...

  

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1. Consider 121 0 0 251 1 0 A= 372 2 -2 493-1 4 Let R(A) column space of A = R(AT) column space of AT, N(A) null space of A, = N(AT) null space of AT. = (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. [10] b (b) Find all the vectors b = b ER such that the system Ax = b is consistent. [6] b3 b [Try to establish a relation in the components of b.] Using this result, decide for -2 what value of ER the system Ax = b is inconsistent where b = -1 [Instruction: Do not solve Ax = b explicitly.] (c) Do you see any relation between the vector b for which Ax = b is consistent and N(AT)? Can you prove this relation in general? [4] 44 A Go 1. Consider 121 0 0 251 1 0 A= 372 2 -2 493-1 4 Let R(A) column space of A = R(AT) column space of AT, N(A) null space of A, = N(AT) null space of AT. = (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. [10] b (b) Find all the vectors b = b ER such that the system Ax = b is consistent. [6] b3 b [Try to establish a relation in the components of b.] Using this result, decide for -2 what value of ER the system Ax = b is inconsistent where b = -1 [Instruction: Do not solve Ax = b explicitly.] (c) Do you see any relation between the vector b for which Ax = b is consistent and N(AT)? Can you prove this relation in general? [4] 44 A Go