I'm currently working on trying to understand column spaces, but I'm struggling to understand one thing. Namely, that if we have a $m × n$ matrix, how does it come that the column space is a subspace of $R^m$. It feels wrong because in the $m × n$ matrix we've actually got n variables and this should mean that we are in $R^n$, right?
Thanks for all help! If there's any good explanations / videos about column spaces, null spaces, rank, and such, I'd be pleased if you could share them with me.
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$\begingroup$As your matrix is of size $m×n$. You can find a linear transformation $T:\Bbb{R}^n\to\Bbb{R}^m$ , $X\mapsto AX$, where $X$ is a column vector of size $n×1$. Column space is a made of all linear combinations of column vectors of a matrix.
Here is a link you can go through this.
$\endgroup$ 0 $\begingroup$The span of any set of vectors is a subspace of the vector space they are a subset of. The columns of the matrix are vectors in $\mathbb{R}^m$ so their span is a subspace of $\mathbb{R}^m$. If you need a proof of this statement I'd be happy to provide, but it's a fun exercise to do on your own as well.
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