Difference between revisions of "Linear map/Definition"

Revision as of 10:33, 12 June 2015 (view source) Alec (Talk | contribs) m ← Older edit		Latest revision as of 10:34, 12 June 2015 (view source) Alec (Talk | contribs) m
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	* <math>T(\lambda x)=\lambda T(x)</math>		* <math>T(\lambda x)=\lambda T(x)</math>
	Or indeed:		Or indeed:
−	* <math>T(x+\lambda y)=T(x)+\lambda T(y)</math>	+	* <math>T(x+\lambda y)=T(x)+\lambda T(y)</math><ref>Linear Algebra via Exterior Products - Sergei Winitzki</ref><noinclude>
		+	==References==
		+	<references/>
		+	</noinclude>

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Latest revision as of 10:34, 12 June 2015

Given two vector spaces $(U,F)$ and $(V,F)$ (it is important that they are over the same field) we say that a map,

$$T:(U,F)\rightarrow(V,F)$$ or simply $$T:U\rightarrow V$$ (because mathematicians are lazy), is a linear map if:

• $$\forall \lambda,\mu\in F$$ and $$\forall x,y\in U$$ we have $$T(\lambda x+\mu y) = \lambda T(x) + \mu T(y)$$

Which is eqivalent to the following:

• $$T(x+y)=T(x)+T(y)$$
• $$T(\lambda x)=\lambda T(x)$$

Or indeed:

• $$T(x+\lambda y)=T(x)+\lambda T(y)$$ ^[1]

References

1. Jump up ↑ Linear Algebra via Exterior Products - Sergei Winitzki