The conversation between human and AI assistant.
What is the physical significance of rank of matrix?
That makes sense. Can the rank of a matrix ever be greater than the number of columns in the matrix?
No, the rank of a matrix can never be greater than the number of columns in the matrix. In fact, the rank can never be greater than the minimum of the number of rows and columns in the matrix. This is because in a matrix with more columns than rows, there must be at least one linearly dependent column. And in a matrix with more rows than columns, there must be at least one row that is a linear combination of the other rows. Therefore, the maximum rank of a matrix is always limited by its size.