Prove linear transformation is invertible

**page929** · September 18th 2011, 10:13 AM

I need to prove that if T is a linear transformation from R^n to R^n, which in onto, then T is invertible.

I am not sure where to begin. Any help would be great.

**zoek** · September 18th 2011, 12:52 PM

Originally Posted by page929

I need to prove that if T is a linear transformation from R^n to R^n, which in onto, then T is invertible.

I am not sure where to begin. Any help would be great.

If $T:V_1\to V_2$ is a linear transformation then:

1. it is invertible $\iff$ it is onto and $1-1$ ,

2. it is $1-1$ $\iff\, Ker(f) = \{O\}$ ,

3. it is onto $\iff Im(f) = V_2$ ,

4. $\text{dim}\, V_1 = \text{dim} \,Ker(f) + \text{dim}\, Im(f)$ .

**HallsofIvy** · September 18th 2011, 04:25 PM

Are you allowed to use the "rank-nullity" theorem, that if A:U->V then the dimension of AU plus the dimension of the kernel of A is equal to the dimension of U? If A is "onto" $R^n$ , its rank is n. Since the A is from $R^n$ , its nullity is n- n= 0 so A is also "one to one".

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