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Let A be a $n \times n$ matrix and $[A_{1},...,A_{n}]$ The columns of the matrix,

my question is:

Why if $Ax=b$ then $A^{T}b\in Span [A_ {1}, ..., A_ {n}]$?

Where x is any vector of length n

Thanks for your help

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    @Bitwise is correct2012-09-27

2 Answers 2

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$A^Tb$ is a vector in the row space of $A$. Since $A$ is square, row space = column space, so $A^Tb$ is also a vector in the column space of $A$.

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Hint: it might be enlightening to write it out by components for $A$ being $1 \times 2$ and again $2 \times 1$

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    @MiguelMoraLuna: it works the same for a square matrix.2012-09-27