## Linear Algebra Question

Forum for the GRE subject test in mathematics.
amateur
Posts: 42
Joined: Wed Sep 10, 2008 9:41 am

### Linear Algebra Question

Question: Let "R" be the reduced row echelon form of an m x n matrix "A". Is the span of the columns of "R", (where each column represents a column vector) equal to the span of the columns of "A"? Justify your answer.

Initially I answered yes, since the operations converting a matrix to its reduced row echelon form are reversible, but the answer at the back of the book is "No".

Any ideas?

P.S. This question is taken from the book "Elementary Linear Algebra", (Here's the URL)

http://www.math.ilstu.edu/matrix/default.html

Thanks.

blp
Posts: 17
Joined: Sun Oct 19, 2008 5:24 am
It's a typo in the book.

Nameless
Posts: 128
Joined: Sun Aug 31, 2008 4:42 pm
I think they are equivalent

amateur
Posts: 42
Joined: Wed Sep 10, 2008 9:41 am

Consider Page 9, 10 of

http://www.ms.uky.edu/~lee/amspekulin/r ... n_ax=b.pdf

for an example.

As a further example, for an m x n matrix, where m is strictly greater than n, and all entries are non-zero, then, in the reduced row echelon form, the last row will necessarily be zero.

So for a 3 x 2 matrix although the column vectors of the original matrix span part of R^3, the column vectors of the reduced row echelon form will only span part of R^2.

Hence the answer to the original question is "Not necessarily", since

[Row Operations may affect the column space of a matrix

ralphhumacho
Posts: 8
Joined: Tue Mar 04, 2008 1:02 am
Simple, easy counterexample (these are the types you should try to think of on the GRE).

A=
| 1 1 |
| 1 1 |

R=
| 1 1 |
| 0 0 |

Clearly, both have different column spaces. Row operations do not necessarily preserve the column space.

Nameless
Posts: 128
Joined: Sun Aug 31, 2008 4:42 pm
Question: Let "R" be the reduced row echelon form of an m x n matrix "A". Is the span of the columns of "R", (where each column represents a column vector) equal to the span of the columns of "A"?
After take another look at your question :

R is the reduced row so the columns may be changed ---->the space spanned by the columns would be changed , therefore they are not equivalent.

Btw, I don't know about you guys, one of my weakness is not to read the problems carefully. This is killing me, hope that you guys don't make stupid mistakes like me while taking the real exams.

blp
Posts: 17
Joined: Sun Oct 19, 2008 5:24 am
Yeah, I'm having the same problem... Another example where I overlooked the fact that we were talking about the span of column spaces, not rows...