Define a left inverse for a matrix and show that TVW has a l

Define a left inverse for a matrix and show that T:V-->W has a left inverse if and only if T is one-to-one.

If A is mxn matrix and rank of A is n then we say that A has left inverse: an nxm matrix B such that

BA = I

Proof:

Assume that T:V->W has left inverse S (say), Now we need to prove that if T(x)=T(y) => x=y

Let T(x)=T(y)

Apply S on both sides, we get:

S(T(x))=S(T(y))

Now since S is left inverse of T, so S(T(x)) = x and also S(T(y)) = y

Hence S(T(x)) = S(T(y)) => x=y

hence T(x)=T(y)=> x=y => T is one to one

Now assume T is one to one, then we need to prove that T has left inverse

Let y in T(V) then y=T(x) for some x in V. Now sinc T is one to one so there is exactly one such x in V.Define this x = S(y)

And so we have whenever S(y) = x means that T(x) = y

And S(T(x)) = S(y) = x => S(T(x)) = x for each x in V and so ST = I. Therefore function S so defined is left inverse of T

Hence proved