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