4.2 Column Vector Addition

#Type/Definition #Topic/Linear_Algebra

Types: N/A
Examples: 4.2.1 Shifting a Vertex Model, 4.2.2 Vector Arithmetic in Mass Spring Chain, 4.2.3 Hooke's Law Data Analysis
Constructions: N/A
Generalizations: N/A

Properties: 4.3 Algebraic Properties of Vector Arithmetic
Sufficiencies: N/A
Questions: N/A

Column Vector Addition

Given column vectors $\vec{x}, \vec{y} \in R^{n}$ , we define addition of these vectors as

\vec{x} + \vec{y} = {[\begin{matrix} x_{1} \\ x_{2} \\ ⋮ \\ x_{n} \end{matrix}]}_{n \times 1} = {[\begin{matrix} y_{1} \\ y_{2} \\ ⋮ \\ y_{n} \end{matrix}]}_{n \times 1} = [\begin{matrix} x_{1} + y_{1} \\ x_{2} + y_{2} \\ ⋮ \\ x_{n} + y_{n} \end{matrix}]

For a single vector-vector addition add, we do $n$ scalar adds. Note that the vector dimensions MUST be identical.

Remark. Notice the sum of two column vectors $\vec{x}, \vec{y} \in R^{n}$ is also a column vector $\vec{x} + \vec{y} \in R^{n}$ . Thus, column vector addition is a 2.3 Relation from $R^{n} \times R^{n}$ to $R^{n}$ . Column vector addition can be used to combine information.
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Remark. When we don't write stuff but we see it anyways:

$\sqrt{16} = 4$
$\log (100) = 2$
$R^{n} = R^{n \times 1}$

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