Let L:R->R be a linear operator with matrix C. Prove if the columns of C are linearly independent, then L is an isomorphism.
The Attempt at a Solution
Assume the columns of C are linearly independent. Then, the homogenous equation Cx=0 is the trivial solution. Need to show L is both 1-1 and onto. Assume that L(c1)=L(c2). Need to show that c1=c2. Well, since the matrix is linearly independent and Cx=0 is trivial solution, then each column represents its own solution. Therefore, if L(c1)=L(c2), then it must be that c1=c2. Now need to show L is onto. So for every d, there exists a c such that L(c)=d. Since C is linearly independent, no vector can be expressed as a linear combination of others. So, for each d, there will be a c where L(c)=d. Therefore, L is an isomorphism.