Problem A: Let A be an m by n matrix with n > m. Explain why the set of column v

Question

Problem A: Let A be an m by n matrix with n > m.
Explain why the set of column vectors of A cannot beindependent.

Problem B: Let A be an m by n matrix with m > n.
Is it possible for the vector space CS(A) to have dimension m?

Problem C: Let A be the 4 by 3 matrix
1 1 0
1 2 1
1 3 0
1 4 2
(i) Give an LU decomposition for A.
(ii) Show the set of three columns vectors of A isindependent. (Partiticularly this question)
(iii) Find a 4 by 1 vector which is not in CS(A).
(iv) Answer (i) and (ii) and (iii) for the 4 by 3 matrix PA inplace of A,
where P is the 4 by 4 matrix
0 0 1 0
0 0 0 1
1 0 0 0
0 1 0 0

Explanation / Answer

by applying the elementary operations on the given matrix, wecan reduce it to normal form. so, row rank = column rank. in the given matrix, the number of rows < no. ofcolumns. assume the maximum possible case . that is all the rowsof the matrix are independent. in such a case also, the matrix can be reduced to the normalform Im . so, there are n-m columns becomingzero during the elementary operations areperformed. that is there are n - m columns of the given matrix aredependent. please post the remaining questions in thenext. thank you.

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