In statistical theory, a common requirement is that a matrix be of full rank. That is, the rank should be as large as possible. Explain why an m×n matrix with more rows than columns has full rank if and only if its columns are linearly independent. Consider the system Ax = 0, where A is an m x n matrix with m > n.
A. Since the rank of A is the number of pivot positions that A has and A is assumed to have full rank, rank A= n. By the Rank Theorem, dim Nul A= n-rank A= 0. So Nul A does not contain only the trivial solution. This happens if and only if the columns of A are linearly independent.
B. Since the rank of A is the number of pivot positions that A has and A is assumed to have full rank, rank A= m. By the Rank Theorem, dim Nul A-m-rank A= 0. So NuIA 3(0), and the system Ax=0 has only the trivial solution. This happens if and only if the columns of A are linearly independent.
C. Since the rank of A is the number of pivot positions that A has and A is assumed to have full rank, rank A= n. By the Rank Theorem, dim Nul A= n-rank A=0. So Nul A3(0), and the system Ax=0 has only the trivial solution. This happens if and only if the columns of A are linearly independent.
D. Since the rank of A is the number of pivot positions that A has and A is assumed to have full rank, rank A= n. By the Rank Theorem, dim Nul A= m-rank A> 0. So Nul A does not contain only the trivial solution. This happens if and only if the columns of A are linearly independent.