Addition on [[Vector|Vectors]] are [[Linear]]: #### Addition $ \huge \let \vec u, \vec v \in \R^{n}$ $\huge \begin{align} \vec u + \vec v = \mat{u_{1}\\u_{2} \\\vdots\\u_{n}} +\mat{v_{1}\\v_{2} \\\vdots\\v_{n}} = \mat{u_{1}+v_{1}\\u_{2}+v_{2} \\\vdots\\u_{n}+v_{n }} \end{align}$ #### Scaling / Multiplication For notation, *never* write a symbol, always put them adjacent, as a $\times$ implies the [[../../02 Areas/Math/Cross Product|Cross Product]] and $\cdot$ implies the [[../../02 Areas/Math/Dot Product|Dot Product]]. $\huge\begin{align*} \vec{v} &= \mat{4\\-1} \\ 2\vec{v} \cdot 2 &= \mat{4\cdot2\\-1\cdot2} \end{align*} $ [[Scalar]] multiplication with multiply the scalar with each component of the vector, resulting in *the length of the vector being scaled*. For any scalar, $k$, and any vector $\vec{v}$, $k\vec{v}$ is vector in the *same* ($k^+$) or *opposite* ($k^-$). *Distributive Property* Applies $\huge{ k(\vec{v} + \vec{u} ) = k\vec{v}+k\vec{u} } $