Column Space of Matrix A and ref(A)

[lkh1986]

Homework Statement

Given a matrix A. So I can reduce A to ref(A). Let's say in ref(A), the columns that contain leading ones are column 1, 3, and 5. True or false:
(a) Columns 1, 3, and 5 from ref(A) form the column space of ref(A).
(b) The corresponding column 1, 3, and 5 from the original matrix A form the column space of matrix A.
(c) Columns 1, 3, and 5 from ref(A) form the column space of the matrix A.
(d) The corresponding column 1, 3, and 5 from the original matrix A form the column space of ref(A).

Homework Equations

The Attempt at a Solution

(a) and (b) are straight forward and hence, both are true. I think (c) is false. Not sure about (d) though.

For (c), I have a specific counter example. I have column space of ref(A) is something like {[1 0 0 0], [0 1 0 0], [0 0 1 0]}, whereas the column space of the original matrix A is {[1 3 2 -1], [-2 2 3 2], [3 1 2 4]}. Notice that the 4th entry for the space spanned by ref(A) will always be 0, but it's possible to have a non-zero value for the space spanned by the column space, if the answers are taken from the original matrix A.

 

[HallsofIvy]

When we say that "vectors a, b, and c form as subspace", we mean that they make span the space. Of course, there may be manysets of vectors that span the same subspace. The important fact here is that the column space of A and the coumn space of ref(A) are the same.