Assum T: R^m to R^n is a matrix transformation with matrix A.Prove that if the column
Assum T: to 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.
If A is a matrix, v and w are vectors and c is a scalar then
Let T be defined as
Further obtain the result as follows:
As the colums of A are linearly idependent, no rows Av will be equal. Thus, the vector obtained is different than one another in
Hence, if the columns of A are linearly independent, then T is one to one.