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Elements

Rank-nullity theorem—The rank-nullity theorem states that for any linear map ${\displaystyle \Phi$ where ${\mathcal {V}}$ is finite-dimensional, the dimension of ${\mathcal {V}}$ equals the sum of the map's rank and nullity.^[1]^[2]^[3]

$\dim {\mathcal {V}}=\dim \operatorname {img} \Phi +\dim \ker \Phi$ $\dim {\mathcal {V}}=\operatorname {rank} \Phi +\operatorname {nullity} \Phi$

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Every linear injection has a left-inverse. $\Phi _{\mathrm {L} }^{-1}=(\Phi ^{\intercal }\Phi )^{-1}\Phi ^{\intercal }$

Every linear surjection has a right-inverse.

$\Phi _{\mathrm {R} }^{-1}=\Phi ^{\intercal }(\Phi \Phi ^{\intercal })^{-1}$

Commentary

There is hardly any theory which is more elementary [than linear algebra], in spite of the fact that generations of professors and textbook writers have obscured its simplicity by preposterous calculations with matrices.

— Jean Dieudonné, Treatise on Analysis, Volume 1

We share a philosophy about linear algebra: we think basis-free, we write basis-free, but when the chips are down we close the office door and compute with matrices like fury.

— Irving Kaplansky, in writing about Paul Halmos

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