35 lines
1012 B
Markdown
35 lines
1012 B
Markdown
#uw/notes #uw/class/math208
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- - -
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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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> [!NOTE] Basic Properties
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> if
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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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**Equality:**
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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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**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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**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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