 Solution (source code)

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= Solution

The \b[<rank-nullity theorem>] states that for a <linear map> $A:X\to Y$ with finite-dimensional domain,
$$
\boxed{\dim X=\dim\ker A+\dim\operatorname{im}A.}
$$
Equivalently, nullity plus rank equals the dimension of the domain. One proof extends a <basis> of $\ker A$ to a <basis> of $X$; the images of the added vectors form a <basis> of $\operatorname{im}A$.

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