What Are Linear Operators?

A linear operator is a linear map from a vector space to itself. Denote the set of all operators on $V$ by $\mathcal{L}(V)$.

Now some examples and non-examples:

Now we have seen some concrete examples, consider the following question: When does an operator invertible? Recall that a linear map is invertible if it is injective and surjective. You may wonder if this is true if there is only injective or surjective. Unfortunely the above third example shows a counterexample for this in infinite-dimensional spaces. But in finite-dimensional spaces, it's true!

Theorem 1. For an operator $T\in \mathcal{L}(V)$, $T$ invertible $\Leftrightarrow$ $T$ injective $\Leftrightarrow$ $T$ surjective.