Section 2.4 Linear combination
We define a linear combination of vectors in a vector space.
Definition 2.4.1.
Let \(V\) be a vector space over a field \(F\text{.}\) A vector \(w\in V\) is said to be a linear combination or an \(F\)-linear combination of vectors \(v_1,v_2,\ldots,v_r\) in \(V\) if there are scalars \(\alpha_1,\alpha_2,\ldots,\alpha_r\in F\) such that
\begin{equation*}
w=\alpha_1 v_1+\alpha_2 v_2+\cdots+\alpha_r v_r.
\end{equation*}
Example 2.4.2.
\begin{equation*}
(\alpha,0)=\alpha(1,0)
\end{equation*}
and
\begin{equation*}
(\alpha,0)=\alpha(1,0)+0(0,1).
\end{equation*}
Remark 2.4.3.
If \(w,z\in V\) are such that
\begin{equation*}
w=\sum_{i=1}^{r}\alpha_iv_i\quad\text{and}\quad z=\sum_{i=1}^{r}\beta_i v_i\quad\text{for }\alpha_i,\beta_i\in F\text{ and }v_i\in V
\end{equation*}
then,
\begin{equation*}
w+z=\sum_{i=1}^{r}(\alpha_i+\beta_i)v_i.
\end{equation*}
Furthermore, for any \(\gamma\in F\)
\begin{equation*}
\gamma w=\sum_{i=1}^{r}(\gamma\alpha_i) v_i\quad\text{and}\quad \gamma z=\sum_{i=1}^{r}(\gamma\beta_i)v_i.
\end{equation*}