John was solving the following problem Prove that the dimens

John was solving the following problem \"Prove that the dimension of the column space of a matrix A is equal to the number of leading 1s in the reduced row echelon form of A.\" His answer was: \"Let A be a matrix. The column space of A is the pivot columns of A (proven in class), and therefore is equal to the number of the leading 1s in the RREF of A.\". Write the corrected solution here.

Solution

Solution:

Let A be a matrix .

The basis for column space of A is the pivot colums of A.

So, dimension of column space is equal to no of pivot columns

We know that leading 1s in RREF occur in pivot colums at pivot positions.

This implies:

Dimension of column space is equal to no. of leading 1s in the RREFof A