9. Let M4x3 be the vector space of all 4 x 3 matrices with real entries. Note that M4x3 = R12 (M4x3 is isomorphic to R12). Let Z4x3 = {A M4x3 | all row and column sums of Z are zero}. For example, A= -5 3 2 1 -3 2 1 2 -3 3 -2 -1 is an element of 24x3. (a) Find a 7 x 12 matrix C whose null space is isomorphic to 24x3. In other words, find a matrix C so that ker CZ4x3 Note that ker C is a subspace of R12, but this subspace is isomorphic to 24x3. (b) Using Matlab, we find that the reduced row echelon form of C has 6 pivot columns (Matlab command: rref (C)). Explain how to use this fact (there are six pivot columns) to find the dimension of Z4x3. (C) What is the dimension of Z4x3? (d) Now that you know the dimension of Z4x3, find a basis for 24x3. (If you find as set of n linearly independent vectors (matrices) in 24x3 where n = dim(Z4x3), it will be a basis.) (Note: do not try to find an orthogonal basis. This is not possible anyway, because no inner product has been defined.) 9. Let M4x3 be the vector space of all 4 x 3 matrices with real entries. Note that M4x3 = R12 (M4x3 is isomorphic to R12). Let Z4x3 = {A M4x3 | all row and column sums of Z are zero}. For example, A= -5 3 2 1 -3 2 1 2 -3 3 -2 -1 is an element of 24x3. (a) Find a 7 x 12 matrix C whose null space is isomorphic to 24x3. In other words, find a matrix C so that ker CZ4x3 Note that ker C is a subspace of R12, but this subspace is isomorphic to 24x3. (b) Using Matlab, we find that the reduced row echelon form of C has 6 pivot columns (Matlab command: rref (C)). Explain how to use this fact (there are six pivot columns) to find the dimension of Z4x3. (C) What is the dimension of Z4x3? (d) Now that you know the dimension of Z4x3, find a basis for 24x3. (If you find as set of n linearly independent vectors (matrices) in 24x3 where n = dim(Z4x3), it will be a basis.) (Note: do not try to find an orthogonal basis. This is not possible anyway, because no inner product has been defined.)