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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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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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.

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

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