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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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    ...col$u$mn space.2011-09-12

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You can check the rank of $\left[ \begin{array}{c|c}A &b \end{array} \right]$ if that is an alternative way that you are looking for (note that row reduced echelon form is also a way of to compute the rank).

In particular, if $A$ is fat i.e. number of rows $m$ is less than the number of columns $n$, then infinitely many solution exists. If $m=n$ then either the matrix is invertible and every $b$ has a solution in the form of $x=A^{-1}b$ or $A$ is singular and you can reduce it to either a fat or tall matrix with full row/column rank.

The last option is the tall matrix case. Then you can simply check if $\operatorname{rank}\left[ \begin{array}{c|c}A &b \end{array} \right]>\operatorname{rank}A$ In case of a positive answer : No $x$ can lead to that particular $b$ (which is usually the case). Hence the popularity of least squares solution.

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    I now see that you asked a question very similar to this answer. So let me know if this is too elementary for you and I can remove it.2011-09-12