Finding a basis for Col(A) when A is not in REF form.

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We previously saw that when A was already in REF, we got a basis by looking at the columns with leading 1s. In this video, we investigate what to do if the matrix A needs to be row reduced first.

This video is part of a Linear Algebra course taught at the University of Cincinnati.

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should the col(A) be the span of the set of the [1 2 3] and [1 3 4] instead of just the set of the two vectors?

wendyzhou

00:02 Finding a basis for Col(A) when A is not in REF form
00:24 Finding a basis for Col(A) when A is not in REF form.
00:45 Column operations change leading ones
01:06 Using row operations to put matrix into row echelon form changes column vectors and may not affect solution set.
01:22 Column space changes if A is not in REF form
01:44 Understanding the basis for column space
02:08 Finding a basis for Col(A) when A is not in REF form
02:24 The original matrix's first and third column forms the basis for the column space.

AyushJayprakashSingh

Excellent video as usual, just some comments.

In general the column space of a matrix A isn't the same as the column space of RREF(A). But, you can always find the column space by using elementary column operations or by taking the transpose of the matrix and then using row reduction to find the row space.

Alternatively you can use the theorem that the column space of a matrix A has as a basis the set of column vectors corresponding to columns with a pivot in RREF(A).

xoppa

1:30 The third column should have a zero above 1, in RREF form.

xoppa

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