Question

Let T: R^n -> R^n be a linear transformation, and A be the matrix of the linear transformation. Prove that if det(A) does not equal 0 then T is one-to-one ?

Answer 1

I know that A must be invertible but not sure how to prove it or tie its not the determinant

Answer 2

*into

Answer 3

if \(T(x)=T(y)\)

then \(Ax=Ay\)

and

\(Ax-Ay=\vec{0}\)

\[A(x-y)=\vec{0}\]

if \(det(A)\neq 0\) then \(A\) is invertable and therefore

\[x-y=\vec{0}\Rightarrow x=y\]

hence \(T\) is one-to-one