# Definition A linear operator on a [[vector space]] $V$ (over a [[field|field]] $\mathbb{F}$) is an [[Operator]] $T$ which obeys [[Linear Map|linearity]], i.e., given $\pmb{v}, \pmb{w} \in V$ and $c \in \mathbb{F}$, we have $ T(c \pmb{v} + \pmb{w}) = c T(\pmb{v}) + T(\pmb{w}) $ # Inverse A linear operator $T$ on $V$ must be a [[Types of Maps|bijection]] from the space $V$ to itself for its inverse $T^{-1}$ to exist. For linear operators, one can show that, if $ T(\pmb{v}) = 0 \implies \pmb{v} = 0 $ then $\exists \, T^{-1}$. To put it in words, $T$ is invertible if and only if the only [[vector]] it sends to $0$ is the zero vector. P.S. [[Types of Maps|surjective]] linear maps are automatically [[Types of Maps|injective]] and vice versa, thus we only need to prove surjectivity or injectivity to prove bijectivity for a linear [[map]]. # Basis A linear operator is not a [[matrix]], just like a [[vector]] is not a column vector. These identifications can only be made once a [[basis]] is chosen, but these objects exist irrespective of [[basis]] choice. Consider a linear operator $T$ on a finite dimensional space $V$, $\pmb{v} \in V$, and choose a basis $\mathcal{B} = \{\pmb{e}_i\}_{\{i = 1, \ldots, n\}}$. The action of $T$ is determined by its action on the basis vectors as follows: $ T(\pmb{v}) = T( \sum_{i = 1}^n v^i \pmb{e}_i) = \sum_{i = 1}^n v^i T(\pmb{e}_i) = \sum_{i, j = 1} v^i T_{i}^{\,j} \pmb{e}_j $ where $ T(\pmb{e}_i) = \sum_{j = 1}^n T_{i}^{\,j} \pmb{e}_j $ the numbers $T_{i}^{\,j}$ are the [[components]] of $T$ in the basis $\mathcal{B}$. The [[matrix]] of $T$ in the basis $\mathcal{B}$, $[T]_\mathcal{B}$, just has components $T_{i}^{\,j}$ where the lower index refers to columns and the upper index refers to rows.The action of the operator in a basis takes the form: $ [T(\pmb{v})]_\mathcal{B} = [T]_\mathcal{B} [\pmb{v}]_\mathcal{B} $ Thus, when we pick a basis, we can use the operator to act on vectors via matrix multiplication. # Vector Space The set of all linear operators on a vector space $V$ forms a [[vector space]] denoted $\mathcal{L}(V)$. For example, we can interpret real $n \times n$ real matrices as linear operators on $\mathbb{R}^n$ ([[Real Coordinate Space|R^n]]) that act on columns vectors via [[matrix]] multiplication. Thus, $M_n(\mathbb{R})$ can be viewed as a vector space whose elements themselves are linear operators. Another example is the quantum mechanical position operator $\hat{x}$ in 1D, which is a linear operator acting on $L^2([-a,a])$ (see [[square-integrable functions]]). Angular momentum operators act on $P_l(\mathbb{R}^3)$ (see [[Polynomial|vector space of polynomials]]).