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 $\displaystyle T:V_1\to V_2$ is a linear transformation then:
1. it is invertible $\displaystyle \iff$ it is onto and $\displaystyle 1-1$,
2. it is $\displaystyle 1-1$ $\displaystyle \iff\, Ker(f) = \{O\}$,
3. it is onto $\displaystyle \iff Im(f) = V_2$,
4. $\displaystyle \text{dim}\, V_1 = \text{dim} \,Ker(f) + \text{dim}\, Im(f)$.
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" $\displaystyle R^n$, its rank is n. Since the A is from $\displaystyle R^n$, its nullity is n- n= 0 so A is also "one to one".