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Why is the nullspace of an $m\times n$ matrix $A$ a subspace of $\mathbb R^n$ whereas the column space is a subspace of $\mathbb R^m$? I understand the dimension of $C(A)$ is designated by the number of components $m$ in each column vector, so the dimension of $N(A)$ is designated by the number of components in each row $n$, but why is the nullspace different like this? |
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Multiplication by an $m\times n$ real matrix $A$ is a linear transformation from $\Bbb R^n$ to $\Bbb R^m$. By definition the nullspace of $A$ is the kernel of that linear transformation, i.e., the set of vectors in the domain of the transformation that are set to the $0$ vector in the range. It is $$\left\{v\in\Bbb R^n:Av=\vec 0\right\}\;.$$ The domain is $\Bbb R^n$, so the nullspace is necessarily (by definition) a subset of $\Bbb R^n$. The column space of $A$ is simply the range (or image) of that linear transformation: it is $$\left\{Av:v\in\Bbb R^n\right\}\;,$$ which must be a subset of $\Bbb R^m$, since every $Av$ with $v\in\Bbb R^n$ is an $m$-place vector. |
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