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Does the column space of a matrix A always equal the column space of the rref(A)? i.e. are the solution sets to Ax=b, or even Ax=0 the same for A and rref(A)?

When doing some examples of matrices that had some linearly independent columns it seemed the Span was preserved by row operations. However, I'm not sure that is the case if the columns are Linearly Dependent. For example, the solution set to the Matrix with columns <1,1> and <2,2> geometrically span a line in 2 space with slope 1, but the rref of said matrix has columns <1,0> and <2,0>, which geometrically Span the x-axis.

Perhaps someone can elaborate on the relationship between a matrix and it's reduced row echelon form, or perhaps point me toward some material that would help me better understand. I have done many searches on google and youtube and have come up short.

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# Relationship between column space of a matrix and rref of matrix

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