Assum T: R^m to R^n is a matrix transformation with matrix A.Prove that if the column

babeeb0oL 2020-11-30 Answered

Assum T:Rm to Rn is a matrix transformation with matrix A. Prove that if the columns of A are linearly independent, then T is one to one (i.e injective). (Hint: Remember that matrix transformations satisfy the linearity properties.
Linearity Properties:
If A is a matrix, v and w are vectors and c is a scalar then
A0=0
A(cv)=cAv
A(v + w)=Av + Aw

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casincal
Answered 2020-12-01 Author has 82 answers
Proof:
Let T be defined as T(v)=Av
Take v=[v1v2..vm]  Rm and A=[a11a12..a1ma21a22..a2m..........an1an2..anm].
Further obtain the result as follows:
T(v)=Av
=[a11a12..a1ma21a22..a2m..........an1an2..anm] [v1v2..vm]
[a11v1+a12v2+ +a1mvma21v1+a22v2+ +a2mvm..........an1v1+an2v2+ +anmvm]
As the colums of A are linearly idependent, no rows Av will be equal. Thus, the vector obtained is different than one another in Rn.
Hence, if the columns of A are linearly independent, then T is one to one.
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