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linear transformations

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Sophie Chrysostomou

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Linear Transformations
Definition: AfunctionT:Rn7−Rmis alinear transformation if forall v,uRn
and for all rR,the following aresatisfied:
i) T(u+v)=T(u)+T(v)
ii) T(rv)=rT(u)
Definition: If T:RnRmis alinear transformation.Then:
1. Rnis the domainof T.
2. Rmis the codomainof T.
3. If WRnthen: the imageofWunderTis
4. the range of Tis T[Rn]={T(v)
5. If WRm,then the inverseimageof Wunder Tis
The set T1[{0}]={vRn
T(v)=)}(where 0Rm)is called the kernelof T.
Example: Let TR37−R2begiven byT([x1,x2,x3]) =[x1x3,x2+x3].
1. ShowthatTis alineartransformation.
2. If H={[x, x,x]
xR},find the imageofHunder T.
3. If U={[1,2],[1,3]},find the inverseimageof Uunder T.
4. Find the kernelof T.
2011 bySophie Chrysostomou
Theorem: Let T:RnRmbealinear transformation, then:
i) if v1,v2,v3,· · · ,vkRnand r1,r2,· · · ,rkRthen
T(r1v1+r2v2+· · · +rkvk)=r1T(v1)+r2T(v2)+· · · +rkT(vk).
ii) T(0)=0where 0Rnand 0Rm.
2011 bySophie Chrysostomou

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Linear Transformations Denition: A function T : R n R m is a linear transformation if for all v,u R n and for all r R, the following are satised: i) T(u + v) = T(u) + T(v) ii) T(rv) = rT(u) Denition: If T : R R m is a linear transformation.Then: n 1. R is the domain of T. 2. R m is the codomain of T. n 3. If W R then: the image of W under T is T[W] = {T(w) w W}. n n 4. the range of T is T[R ] = {T(v) v R } . 5. If W R , then the inverse image of W under T is T 1[W ] = {v R n T(v) W } 1 n m The set T [{0 }] = {v R T(v) = )} ( where 0 R ) is called the kernel of T.
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