Let $V$, $W$ be vector spaces over a field $F$. A function $f : V \to W$ is linear if:
- (Additivity) $f(x + y) = f(x) + f(y)$ for all $x$, $y \in V$.
- (Homogeneity) $f(\lambda x) = \lambda f(x)$ for all $x \in V$ and $\lambda \in F$.
Note. Linear function is also known as vector space homomorphism.
Let $V$, $W$ vector spaces over a field $F$ and $f : V \to W$ be a linear mapping. The kernel of $f$ is
\[\ker f = f^{-1}[\{0\}].\]
Let $V$, $W$ be vector spaces over a field $F$ and $f : V \to W$. $f$ is linear if
\[f(\lambda x + y) = \lambda f(x) + f(y)\]for all $x$, $y \in V$ and $\lambda \in F$.
Let $V$, $W$ be vector spaces. For any linear mapping $f : V \to W$:
- $f(0) = 0$.
- $f(x - y) = f(x) - f(y)$ for all $x$, $y \in V$.
Let $V$, $W$ be vector spaces over a field $F$ and $f : V \to W$ be a linear mapping. For any linear dependent $S \subseteq V$, $f[S]$ is linear dependent.
Let $V$, $W$ be vector spaces over a field $F$ and $X \subseteq V$, $Y \subseteq W$ be subspaces. For any linear mapping $f : V \to W$, $f[X]$ and $f^{-1}[Y]$ are subspaces.
Let $V$, $W$ be vector spaces over a field $F$. For any linear mapping $f : V \to W$:
- $\dim \im f \leq \dim V$.
- If $f$ is an isomorphism, then $\dim V = \dim W$.
Let $V$, $W$ be vector spaces over a field $F$. $\hom(V, W)$ is a subspace of $\fun(V, W)$.
For any vector space $V$, $(\endo V, \circ, +)$ is a ring.
Let $V$, $W$ be vector spaces over a field $F$. For any linear mapping $f : V \to W$:
- $\im f$ and $\ker f$ are subspaces.
- $f$ is an isomorphism if and only if $\ker f = \{0\}$.
- If $f$ is an isomorphism, then for any linear independent $S \subseteq V$, $f[S]$ is linear independent.