Theorem. For every linear operator f in R^{n} with standard matrix A the following conditions are equivalent:
Proof. Let us translate the third and the fourth
conditions of this theorem into the language of matrices.
The third condition (f is surjective) means that for every vector
b in R^{n} there exists a vector v in R^{n} such that Av=b,
that is for every vector b the system of equations Av=b has a solution.
The fourth condition (f is injective) means that for every two
vectors x and y
We can rewrite this condition in the
following form:
Let us denote x-y by v. Since u and v were arbitrary vectors, v is also an
arbitrary vector. Then we can write our condition in the
following form:
that is the homogeneous system of linear equations Av=0 has only the trivial solution.
Thus we can rewrite our theorem in as follows:
Theorem. For every linear operator f in R^{n} with standard matrix A the following conditions are equivalent:
Now it is easy to see that the first two properties in this theorem are equivalent because of the theorem about a connection between invertible matrices and invertible operators. The last 3 conditions are equivalent by the second theorem about inverses. Thus all four conditions are equivalent. The theorem is proved.