Suppose A is an m times n matrix Which of the subspaces RowA

Suppose A is an m times n matrix. Which of the subspaces Row(A), Col(A), Null(A), Row(A\'), Col(A\'), and Null(A^T) are in R^m and which are in R^n? How many distinct subspaces are in this list?

Solution

If A is a mx n matrix, then the rows are n- vectors and the columns are m- vectors. Therefore, Row(A) is in Rⁿ and Col(A) is in R^m.

Similarly, since A^T is a nx m matrix, hence the rows of A^T are m- vectors and the columns of   A^T are   n- vectors. Therefore, Row(A^T) is in R^m and Col(A^T ) is in Rⁿ.

We know that Null(A) is the solution space of the equation AX = 0. If A is a m x n matrix, then X has to be set of n-vectors. Hence Null(A) is in Rⁿ. Similarly, A^T is a n x m matrix so that the solution space of A^T X = 0 has to be a set of m-vectors. Hence Null(A^T) is in R^m.

Thus, Col(A), Row(A^T) , Null(A^T) are in R^m and Row(A), Col(A^T) and Null(A) are in Rⁿ.

Of the above subspaces, Row(A) = Col(A^T) and Col(A) = Row(A^T). Hence the distinct subspaces are Row(A), Col(A), Null(A) and Null(A^T).