1.8 KiB
#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 ifu_{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 $$\mathbf{x} = \begin{bmatrix} x_{1} \ x_{2} \ x_{3} \end{bmatrix} = \begin{bmatrix} 2 \ 2 \ 0 \end{bmatrix} + s_{1}\begin{bmatrix} -11 \ -4 \ 1 \end{bmatrix}$$