Consider on 8by12 matrix of rank 7 What are the dimensions o

Consider on 8-by-12 matrix of rank 7. What are the dimensions of the four fundamental subspaces associated with this matrix? What is the sum of those four dimensions?

Solution

Let A be the given 8 x 12 matrix of rank 7.

We know that the 4 fundamental subspaces associated with A are the column space of A (Col (A) ) the column space of A^T (Col(A^T), which is also the row space of A) ,the null space of A ( null(A) and A^T (null(A^T)). The rank of the matrix being 7, the number of non-zero rows in the RREF of A is 7. Then the dimension of Col(A) = rank = 7 = dimension of the row space = Col(A^T) since the row space and column space of a matrix have the same dimension which is equal to the rank of the matrix.

The rank of A^T is equal to dim(Col(A^T) = dimension of the row space of A = 7. The dimension of its null space ( set of solutions of the equation AX = 0) is called the nullity of A. It is denoted null (A)

Now, as per the rank –nullity theorem, the rank and the nullity of a matrix add up to the number of columns of the matrix. Therefore, the dimension of null(A) = 12-7 = 5. Also, the dimension of null(A^T) = 8- 7 = 1. The sum of dimensions of Col(A), Col(A^T), null(A) and null(A^T) is 7+7+5+1= 20