56 lines
1.8 KiB
Markdown
56 lines
1.8 KiB
Markdown
#uw/notes #uw/class/math208
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Vectors can be represented vertically in a matrix (column vector):
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$$\mathbf{u} = \begin{bmatrix}
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u_{1} \\ u_{2} \\ \vdots \\ u_{n}
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\end{bmatrix}$$
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> [!INFO] Basic Properties
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> if $\mathbf{u}$ and $\mathbf{v}$ are vectors defined as
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> $$\mathbf{u} =\begin{bmatrix}
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> u_{1} \\ u_{2} \\ \vdots \\ u_{n}
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> \end{bmatrix} \quad \text{and} \quad \mathbf{v} =\begin{bmatrix}
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> v_{1} \\ v_{2} \\ \vdots \\ v_{n}
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> \end{bmatrix}$$
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>
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> **Equality:**
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>
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> $\mathbf{u}=\mathbf{v}$ if and only if $u_{1}=v_{1}, u_{2}=v_{2},\dots,u_{n}=v_{n}$
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>
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> **Addition:**
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> $$\mathbf{u}+\mathbf{v} = \begin{bmatrix}
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> u_{1} \\ u_{2} \\ \vdots \\ u_{n}
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> \end{bmatrix} + \begin{bmatrix}
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> v_{1} \\ v_{2} \\ \vdots \\ v_{n}
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> \end{bmatrix} = \begin{bmatrix}
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> u_{1}+v_{1} \\ u_{2}+v_{2} \\ \vdots \\ u_{n}+v_{n}
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> \end{bmatrix}$$
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>
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> **Scalar Multiplication:**
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> $$c \mathbf{u} = c \begin{bmatrix}
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> u_{1} \\ u_{2} \\ \vdots \\ u_{n}
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> \end{bmatrix} = \begin{bmatrix}
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> c \cdot u_{1} \\ c \cdot u_{2} \\ \vdots \\ c \cdot u_{n}
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> \end{bmatrix}$$
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A linear combination of vectors is when vectors multiplied by scalar coefficients are added or subtracted
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$$c_{1}\mathbf{u_{1}}+c_{2}\mathbf{u_{2}}+\dots+c_{m}\mathbf{u_{m}}$$
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is a linear combination if all $c$ values are scalars and $\mathbf{u}$ are vectors
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The general form of a solution to a linear system of equations can be represented as a linear combination of vectors
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> [!EXAMPLE] Example
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> if the solution of a linear system of equations is
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> $$\begin{gather} x_{1} = 2 - 11s_{1} \\ x_{2} = 2 - 4s_{1} \\ x_{3} = 0 + 1s_{1} \end{gather}$$ then the vector form of the general solution is
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> $$\mathbf{x} = \begin{bmatrix}
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> x_{1} \\ x_{2} \\ x_{3}
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> \end{bmatrix} = \begin{bmatrix}
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> 2 \\ 2 \\ 0
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> \end{bmatrix} + s_{1}\begin{bmatrix}
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> -11 \\ -4 \\ 1
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> \end{bmatrix}$$
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