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College/AU 26/MATH 208 (Matrix Algebra)/Chapter 2.1 Notes.md
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@@ -7,28 +7,38 @@ u_{1} \\ u_{2} \\ \vdots \\ u_{n}
> [!NOTE] Basic Properties
> if
> 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:**
> $$c \mathbf{u} = c \begin{bmatrix}
> u_{1} \\ u_{2} \\ \vdots \\ u_{n}
> \end{bmatrix} = \begin{bmatrix}
> c \cdot u_{1} \\ c \cdot u_{2} \\ \vdots \\ c \cdot u_{n}
> \end{bmatrix}$$
if $\mathbf{u}$ and $\mathbf{v}$ are vectors defined as
$$\mathbf{u} =\begin{bmatrix}
u_{1} \\ u_{2} \\ \vdots \\ u_{n}
\end{bmatrix} \quad \text{and} \quad \mathbf{v} =\begin{bmatrix}
v_{1} \\ v_{2} \\ \vdots \\ v_{n}
\end{bmatrix}$$
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
**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:**
$$c \mathbf{u} = c \begin{bmatrix}
u_{1} \\ u_{2} \\ \vdots \\ u_{n}
\end{bmatrix} = \begin{bmatrix}
c \cdot u_{1} \\ c \cdot u_{2} \\ \vdots \\ c \cdot u_{n}
\end{bmatrix}$$
The general form of a solution to a linear system of equations can be represented as a linear combination of vectors
if the solution of a linear system of equations is
$$$$