Linear transformaton and inverse

[autre]

Homework Statement

If T : V → W is an injective linear transformation, then T^-1: V →W is a linear transformation.

The Attempt at a Solution

Let w1, w2 in W. If w1=T(v1) and w2=T(v2), v1=/=v2 in V. Thus, T^-1: V →W is a function. Then, v1+v2=T^-1(w1) + T^-1(w2) and for a in F, T^-1(w1) = aT^-1(w1) = av1 for w in W.

 

[micromass]

If T : V → W is an injective linear transformation, then T^-1: V →W is a linear transformation.

This is not true. Injectivity is not enough. You need bijectivity.

 

[autre]

Oh sorry, I left out a part. T:V -> W is a linear transformation, T(V) is a subspace of W.

 

[mtayab1994]

This is not true. Injectivity is not enough. You need bijectivity.

but doesn't he have to use injectivity and surjectivity to show bijectivity?

 

[micromass]

Oh sorry, I left out a part. T:V -> W is a linear transformation, T(V) is a subspace of W.

Not enough. You need T to be surjective.

but doesn't he have to use injectivity and surjectivity to show bijectivity?

Yes, but he didn't state that T is surjective.

 

[Deveno]

what IS true, is that if T:V→W is an injective linear transformation, then

T^-1:T(V)→V is a linear transformation.

now, prove this by explicitly defining what T^-1 has to be.