Linear Function

Definition 1. Let $V$, $W$ be vector spaces over a field $F$. A linear function $f : V \to W$ is a vector space homomorphism:

1. (Additivity) For all $x$, $y \in V$,

   \[f(x + y) = f(x) + f(y).\]

2. (Homogeneity) For all $x \in V$ and $\lambda \in F$,

   \[f(\lambda x) = \lambda f(x).\]

Definition 2. Let $V$, $W$ vector spaces over a field $F$ and $f : V \to W$ be a linear function. The kernel of $f$ is

\[\ker f = f^{-1}[\{0\}].\]

Proposition 1. 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 $\lambda \in F$.

Proposition 2. Let $V$, $W$ be vector spaces. For any linear function $f : V \to W$:

1. \[f(0) = 0.\]

2. For all $x$, $y \in V$,

   \[f(x - y) = f(x) - f(y).\]

Proposition 3. Let $V$, $W$ be vector spaces over a field $F$ and $f : V \to W$ be a linear function. For any linear dependent $S \subseteq V$, $f[S]$ is linear dependent.

Proposition 4. Let $V$, $W$ be vector spaces over a field $F$ and $X \subseteq V$, $Y \subseteq W$ be subspaces. For any linear function $f : V \to W$, $f[X]$ and $f^{-1}[Y]$ are subspaces.

Proposition 5. Let $V$, $W$ be vector spaces over a field $F$. For any linear function $f : V \to W$:

1. \[\dim \im f \leq \dim V.\]

2. If $f$ is bijective, then

   \[\dim V = \dim W.\]

Proposition 6. For any two vector spaces $V$ and $W$, $\hom(V, W)$ is a subspace of $\fun(V, W)$.

Proposition 7. For any vector space $V$, $(\endo V, \circ, +)$ is a ring.

Proposition 8. Let $V$, $W$ be vector spaces over a field $F$. For any linear function $f : V \to W$:

1. $\im f$ and $\ker f$ are subspaces.
2. $f$ is bijective if and only if $\ker f = \{0\}$.
3. If $f$ is bijective, then for any linear independent $S \subseteq V$, $f[S]$ is linear independent.