06 Linear Combinations

If $\vec{v_{1}}, \vec{v_{2}}, \dots , \vec{v_{k}} \in \mathbb{R}^n$ and $c_{1}, c_{2}, \dots, c_{k} \in \mathbb{R}$, then

$$ c_{1}\vec{v_{1}}, c_{2}\vec{v_{2}}, \dots, c_{k}\vec{v_k} $$ is a linear combination of the $\vec{v}$'s using $c$'s as weights.

A linear combination is just a scaled column vector that can be written in terms of other vectors

Example:

$$ \begin{aligned} c_{1} \begin{bmatrix} 1 \\ 4 \\ 7 \end{bmatrix} +c_{2} \begin{bmatrix} 2 \\ 5 \\ 8 \end{bmatrix} +c_{3} \begin{bmatrix} 3 \\ 6 \\ 9 \end{bmatrix} &= \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \\ \\ \implies c_{1}+2c_{2}+3c_{3}&=1 \\ 4c_{2}+5c_{2}+6c_{3} & =1 \\ 7c_{1}+8c_{2}+9c_{3} & =1 \end{aligned} $$

If $\begin{bmatrix}1\\1\\1\end{bmatrix}$ can be be written in terms of $\begin{bmatrix}1\\4\\7\end{bmatrix}, \begin{bmatrix}2\\5\\8\end{bmatrix},$ and $\begin{bmatrix}3\\6\\9\end{bmatrix},$ then it is a linear combination of the 3 aforementioned matrices.