Skip to main content

Linear Algebra: Companion notes

Abhay Soman

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*}
Consider \(\R^2\) as a vector space over \(\R\) (refer to Example 2.2.2). A vector \((\alpha,0)\in\R^2\) is a linear combination of vector \((1,0)\) as well as a linear combination of vectors \((1,0)\) and \((0,1)\text{.}\) Indeed, we have
\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*}
What are all vectors in \(\R^2\) that can be written as linear combinations of vectors \((1,0)\) and \((0,1)\text{?}\) Think geometrically.
<Prev^TopNext>