# Linear transformaton and inverse

1. Dec 3, 2011

### autre

1. The problem statement, all variables and given/known data

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

3. 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.

2. Dec 3, 2011

### micromass

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

3. Dec 3, 2011

### autre

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

4. Dec 3, 2011

### mtayab1994

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

5. Dec 3, 2011

### micromass

Not enough. You need T to be surjective.

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

6. Dec 4, 2011

### 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.

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