Assume T: R∧m to R∧n 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 the 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=0A(cv)=cAv

Question

Saul Murray · Accepted Answer

Step 1Given,T:Rm→Rn&amp;nbsp;is a matrix fransformation wilh matrix A.Since, a matrix transformation is one to one to one if T(x)=b has at mostone solution for every b.Now given columns of matrix A are linearly independ.Then, colymns of A span&amp;nbsp;Rn⇒&amp;nbsp;The equation Ax=b has at least one solution for every b in&amp;nbsp;Rn.⇒&amp;nbsp;The equation Tx=b has at least one solution for every b&amp;nbsp;Rn.Step 2Also use the property that if columns of matrix T are linearlyindepent, thenNullity(T)=0Now take Tx, Ty&amp;nbsp;∈Rn&amp;nbsp;,then use the property of transformationTx=Ty;for some x,y&amp;nbsp;∈Rm⇒&amp;nbsp;Tx-Ty=0⇒&amp;nbsp;Tx+(-1)Ty=0⇒&amp;nbsp;Tx=T(-y)=0&amp;nbsp;[∵T&amp;nbsp;is&amp;nbsp;linear]⇒&amp;nbsp;T(x-y)=0⇒&amp;nbsp;(x-y)&amp;nbsp;∈Nullity(T)⇒&amp;nbsp;(x-y)=0⇒&amp;nbsp;x=yTherefore,Tx=Ty&amp;nbsp;⇒&amp;nbsp;x=yThis implies thet, T is one to one transformation.

Assume T: \(\displaystyle{R}\wedge{m}\) to \(\displaystyle{R}\wedge{n}\) is

Answered question

Answer & Explanation

New Questions in High school geometry