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Say we want to determine the range of the transformation A x. And we call the solution vectors b. So if we want to determine all possible b's, are we going to have to put A into reduce row echelon form? or is there another way?

Or alternatively, given a transformation A and a b, is there a way to tell whether b is in the range without reducing Ax=b into reduced row echelon form?

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    What does "we call the solution $\mathbf{b}$" mean? The range of a linear transformation $A$ is a subspace of a certain vector space. And $A\mathbf{x}$ is not generally a transformation, but the value of the transformation $A$ at the vector $\mathbf{x}$. What exactly are you trying to ask? Are you asking, given a transformation $A$ and a $\mathbf{b}$, is there a way to tell whether $\mathbf{b}$ is in the range without reducing $A\mathbf{x}=\mathbf{b}$ into reduced row echelon form?2011-09-11
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    essentially, yes that is what I am asking.2011-09-11
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    Depends on $A$. Sometimes it is clear what the range is by other considerations; often it is not. The *standard* way to determine the range of a linear transformation is to try to compute the span of $A\mathbf{e}_1,\ldots,A\mathbf{e}_n$, where $\mathbf{e}_1,\ldots,\mathbf{e}_n$ are the standard basis vectors; but that amounts to *precisely* trying to figure out the columnspace of $A$, and the simplest way of doing that is to perform at least *some* row-reduction on $A$ so that the columnspace is "obvious". You often don't have to go *all* the way to reduced row echelon form.2011-09-11
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    I see, so no shortcuts really. Thank you very much for your help.2011-09-11
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    @weeztbizzle: Note that elementary row operations do not respect the column space; that's why you usually work with the augmented matrix. If you want to find a nice basis for the column space, you can row-reduce the *transpose* of $A$, since elementary row operations do respect the rowspace, and the rowspace of the transpose is (essentially) the columnspace of the original.2011-09-11
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    Although elementary row operations do not respect the column space, they do respect linear dependence relations among the columns; columns 2, 4, and 7 (say) are linearly dependent after you do an elementary row operation if and only if those columns were linearly dependent before the operation. So you can go to row echelon form (you don't have to go all the way to reduced rwo echelon form), pick out the columns that form a basis for the column space in the echelon matrix (namely, those with a leading non-zero entry), and the corresponding columns in the original matrix form a basis for its...2011-09-12
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    ...column space.2011-09-12

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