Suppose that Tx Mx Assume that the reduced row echelon form

Suppose that T(x) = Mx. Assume that the reduced row echelon form of M has a pivot in every column. Prove that T is injective. T : R^n right arrow R^m is linear and is given by

Solution

Since the reduced row echelon form of M has a pivot in every column so M is injective

It means if Mx=0 then x=0

now suppose Tx=0 and we have to prove that x=0

Tx=0

thus MXx=0

since M is injective so

X(x)=0

so x=0

therefore T is injective