Linear Maps

Let $L: \mathbb{F}^{n} \rightarrow \mathbb{F}^{m}$ and $T: \mathbb{F}^{n} \rightarrow \mathbb{F}^{m}$ be linear maps with associated matrices $M_{L}~\in~\mathbb{M}_{mn}(\mathbb{F})$ and $M_{T}~\in~\mathbb{M}_{mn}(\mathbb{F})$, and $\bm{x} \in \mathbb{F}^{n}$.

The addition of $L$ and $T$ is defined as

where $L+T$ is also a linear map.

On the associated matrices $M_{L}$ and $M_{T}$ with elements $m_{ij}$ and $t_{ij}$ respectively, this becomes

Let $L: \mathbb{F}^{n} \rightarrow \mathbb{F}^{m}$ and $R,T: \mathbb{F}^{m} \rightarrow \mathbb{F}^{k}$ be linear maps.

Let $L,R: \mathbb{F}^{n} \rightarrow \mathbb{F}^{m}$ and $T: \mathbb{F}^{m} \rightarrow \mathbb{F}^{k}$ be linear maps.

Let $L: \mathbb{F}^{n} \rightarrow \mathbb{F}^{m}$, $R: \mathbb{F}^{m} \rightarrow \mathbb{F}^{k}$ and $T: \mathbb{F}^{k} \rightarrow \mathbb{F}^{q}$ be linear maps.

Let $L: \mathbb{F}^{n} \rightarrow \mathbb{F}^{m}$ and $R: \mathbb{F}^{m} \rightarrow \mathbb{F}^{k}$ be linear maps with associated matrices $M_{L}~\in~\mathbb{M}_{mn}(\mathbb{F})$ and $M_{R}~\in~\mathbb{M}_{km}(\mathbb{F})$.

If $T = R \circ L$ is a linear map then $M_{T} = M_{R}M_{L}~\in~\mathbb{M}_{kn}(\mathbb{F})$.

Let $L: \mathbb{F}^{n} \rightarrow \mathbb{F}^{m}$ be a linear map.

The image of L is defined to be

These are the possible output vectors or the linear map.

The kernel of L is defined to be

These are the input vectors for which the linear map returns the zero vector.

Let $L: \mathbb{F}^{n} \rightarrow \mathbb{F}^{m}$ be a linear map.

$L$ is injective if and only if

where $\bm{0}$ is the zero vector.

This means the only input vector the map can output $\bm{0}$ for is $\bm{0}$.

Let $L: \mathbb{F}^{n} \rightarrow \mathbb{F}^{m}$ be a linear map.

$L$ is bijective if it is both surjective and injective.

Properties:

• Let $L: \mathbb{F}^{n} \rightarrow \mathbb{F}^{n}$ be a linear map and bijective, $L^{-1}: \mathbb{F}^{n} \rightarrow \mathbb{F}^{n}$ be also a linear map, where $L^{-1}$ is the inverse map.

Let $L: \mathbb{F}^{n} \rightarrow \mathbb{F}^{m}$ be a linear map.

The nullity of $L$ is defined to be

where $\textrm{dim}$ is the dimension.

Let $L: \mathbb{F}^{n} \rightarrow \mathbb{F}^{m}$ be a linear map.

The rank of $L$ is defined to be

where $\textrm{dim}$ is the dimension.