Vector Combinations

We can multiply vectors by scalars, add, and subtract. Given vectors $\vec{u},\vec{v},\vec{w}$, we can form the linear combination $x_1\vec{u}+x_2\vec{v}+x_3\vec{w}=\vec{b}$.

An example in ${\mathbb{R}}^3$ would be:

$$\vec{u}=\left[\begin{array}{c}1\\ -1\\ 0\end{array}\right],\vec{v}=\left[\begin{array}{c}0\\ 1\\ -1\end{array}\right],\vec{w}=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$$

• The collection of all multiples of $\vec{u}$ forms a line through the origin.
• The collection of all multiples of $\vec{v}$ forms another line.
• The collection of all combinations of $\vec{u}$ and $\vec{v}$ forms a plane.
• Taking all combinations of some vectors creates a subspace.