Linear Mapping

LM#DEF. Linear Mapping.

Let $V$, $W$ be vector spaces over a field $F$. A mapping $f:V\to W$ is linear if:

1. $f(x + y) = f(x) + f(y)$ for all $x$, $y\in V$ (additivity).
2. $f(\lambda x) = \lambda f(x)$ for all $x\in V$ and $\lambda\in F$ (homogeneity).

• Synonym of linear mapping: vector space homomorphism

LM#DEF-KER. Kernel.

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

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

LM#PROP-TST.

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 $\lambda \in F$.

LM#PROP-A.

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

1. $f(0) = 0$.
2. $f(x - y) = f(x) - f(y)$ for all $x$, $y\in V$.

LM#PROP-LI.

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.

LM#PROP-S.

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.

LM#PROP-DIM.

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

1. $\dim\im f \leq \dim V$.
2. If $f$ is an isomorphism, then $\dim V = \dim W$.

LM#PROP-HOM.

Let $V$, $W$ be vector spaces over a field $F$. $\hom(V,W)$ is a subspace of $\map(V,W)$.

LM#PROP-END.

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

LM#PROP-K.

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

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

Notation

$\map(X,Y)$

The set of all mappings from $X$ to $Y$.