User:SirMeowMeow

{{Math theorem

| name = Rank-nullity theorem

| math_statement = The rank-nullity theorem states that for any linear map $\backslash Phi\; :\; \backslash mathcal\{V\}\; \backslash to\; \backslash mathcal\{W\}$ where $\backslash mathcal\{V\}$ is finite-dimensional, the dimension of $\backslash mathcal\{V\}$ equals the sum of the map's rank and nullity.{{harvtxt|Axler|2015}} p. 63, § 3.22{{harvtxt|Katznelson|Katznelson|2008}} p. 52, § 2.5.1{{harvtxt|Valenza|1993}} p. 71, § 4.3

$$\backslash dim\; \backslash mathcal\{V\}\; =\; \backslash dim\; \backslash operatorname\{img\}\; \backslash Phi\; +\; \backslash dim\; \backslash ker\; \backslash Phi$$

$$\backslash dim\; \backslash mathcal\{V\}\; =\; \backslash operatorname\{rank\}\; \backslash Phi\; +\; \backslash operatorname\{nullity\}\; \backslash Phi$$

}}

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|Observations

• One
• Two
• Three

Every linear injection has a left-inverse.

$$\backslash Phi\_\backslash mathrm\{L\}^\{-1\}\; =\; (\backslash Phi^\backslash intercal\; \backslash Phi)^\{-1\}\; \backslash Phi^\backslash intercal$$

Every linear surjection has a right-inverse.

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

{{Quote box

| quote = 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.

| author = — Jean Dieudonné

| source = Treatise on Analysis, Volume 1

| align = center

}}

{{Quote box

| quote = 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.

| author = — Irving Kaplansky, in writing about Paul Halmos

| align = center

}}

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• {{Cite book|last=Halmos|first=Paul Richard|title=Finite-Dimensional Vector Spaces|publisher= Springer|year=1974|isbn=0-387-90093-4|edition=2nd|author-link=Paul Halmos|orig-year=1958}}

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• {{Cite book|last=Roman|first=Steven|title=Advanced Linear Algebra|publisher=Springer|year=2005|isbn=0-387-24766-1|edition=2nd|author-link=Steven Roman}}

• {{Cite book|last=Rotman|first=Joseph Jonah|title=An Introduction to the Theory of Groups|publisher= Springer|year=1999|isbn=3-540-94285-8|edition=4th|author-link=Joseph J. Rotman}}

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• {{Cite book|last=Trefethen|first=Lloyd Nicholas|title=Numerical Linear Algebra|last2=Bau III|first2=David|publisher=SIAM|year=1997|isbn=978-0-898713-61-9|author-link=Nicholas Trefethen}}

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