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I have several conceptual questions that have been confusing me for a while in linear algebra.

Let A be a 3 by 5 matrix with full rank rows. A is now simplified to Echelon form U and further simplified to row reduced form R. The question is

A) are the column spaces of U, R, and A all equal? Why or why not? (it is trivial to mention their dimensions' equality)

B) are the row spaces of U, R, and A all equal? Why or why not?

Thanks in advance.

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You arrive at these forms by multiplying by elementary matrices... Think about what it does. This would answer your question about row spaces. –  user21436 Apr 24 '12 at 7:17
    
I see. The new rows are linear combinations of the original rows. So the row space should remain constant. And that means the column space dont. Thanks! But what is the relationship among those three column spaces? How are they related geometrically? –  user803253 Apr 24 '12 at 7:22
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The most important fact about the columns of $U$, $R$, and $A$ is that any set of columns of one of them is linearly independent if and only if the corresponding set of columns of each of the other two is linearly independent. This is true because elementary row operations don't affect dependence relations among the columns of a matrix.

In particular, if you have a set of columns forming a basis for the column space of one of the matrices, then the corresponding columns form a basis for the column space of each of the others. And, of course, it's easy to pick out a basis for the column space of $R$.

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