#uw/notes #uw/class/math208 
- - -
Vectors can be represented vertically in a matrix (column vector):
$$\mathbf{u} = \begin{bmatrix}
u_{1} \\ u_{2} \\ \vdots \\ u_{n}
\end{bmatrix}$$


> [!INFO] Basic Properties
> if $\mathbf{u}$ and $\mathbf{v}$ are vectors defined as 
> $$\mathbf{u} =\begin{bmatrix}
> u_{1} \\ u_{2} \\ \vdots \\ u_{n}
> \end{bmatrix} \quad \text{and} \quad \mathbf{v} =\begin{bmatrix}
> v_{1} \\ v_{2} \\ \vdots \\ v_{n}
> \end{bmatrix}$$
> 
> **Equality:**
> 
> $\mathbf{u}=\mathbf{v}$ if and only if $u_{1}=v_{1}, u_{2}=v_{2},\dots,u_{n}=v_{n}$
> 
> **Addition:**
> $$\mathbf{u}+\mathbf{v} = \begin{bmatrix}
> u_{1} \\ u_{2} \\ \vdots \\ u_{n}
> \end{bmatrix} + \begin{bmatrix}
> v_{1} \\ v_{2} \\ \vdots \\ v_{n}
> \end{bmatrix} = \begin{bmatrix}
> u_{1}+v_{1} \\ u_{2}+v_{2} \\ \vdots \\ u_{n}+v_{n}
> \end{bmatrix}$$
> 
> **Scalar Multiplication:**
> $$c \mathbf{u} = c \begin{bmatrix}
> u_{1} \\ u_{2} \\ \vdots \\ u_{n}
> \end{bmatrix} = \begin{bmatrix}
> c \cdot u_{1} \\ c \cdot u_{2} \\ \vdots \\ c \cdot u_{n}
> \end{bmatrix}$$


A linear combination of vectors is when vectors multiplied by scalar coefficients are added or subtracted
$$c_{1}\mathbf{u_{1}}+c_{2}\mathbf{u_{2}}+\dots+c_{m}\mathbf{u_{m}}$$
is a linear combination if all $c$ values are scalars and $\mathbf{u}$ are vectors


The general form of a solution to a linear system of equations can be represented as a linear combination of vectors

> [!EXAMPLE] Example
> if the solution of a linear system of equations is 
> $$\begin{gather} x_{1} = 2 - 11s_{1} \\ x_{2} = 2 - 4s_{1} \\ x_{3} = 0 + 1s_{1} \end{gather}$$ then the vector form of the general solution is 
> $$\mathbf{x} = \begin{bmatrix}
> x_{1} \\ x_{2} \\ x_{3}
> \end{bmatrix} = \begin{bmatrix}
> 2 \\ 2 \\ 0
> \end{bmatrix} + s_{1}\begin{bmatrix}
> -11 \\ -4 \\ 1
> \end{bmatrix}$$