4.1 Scalar-Vector Multiplication

Types: N/A
Examples: 4.1.1 Newton's Second Law & Spring Forces
Constructions: N/A
Generalizations: N/A

Properties: 4.1.2 Geometry of Scalar-Vector Multiplication, 4.3 Algebraic Properties of Vector Arithmetic
Sufficiencies: N/A
Questions: N/A

Scalar-Vector Multiplication

Let $n \in N$ . Let $a \in R$ (scalar value) and $\vec{x} \in R^{n \times 1}$ be a given 3.1 Column Vector. Then, scalar-vector multiplication is defined as the vector

a \cdot \vec{x} = a \cdot [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ ⋮ \\ x_{n} \end{matrix}] = [\begin{matrix} a \cdot x_{1} \\ a \cdot x_{2} \\ a \cdot x_{3} \\ ⋮ \\ a \cdot x_{n} \end{matrix}]

where the $i$ th coefficient of this product is given by $a x_{i}$ for all $i \in {1, 2, \dots, n}$ .

Remark. The definition above is specifically scalar column-vector multiplication. Doing scalar-vector multiplication with a 3.4 Row Vector instead would like:

\begin{aligned} a_{1 \times 1} \cdot {\vec{y}}_{1 \times n} & = a \cdot [\begin{array}{c} y_{1} & y_{2} & y_{3} & \dots & y_{n} \end{array}] \\ = [\begin{array}{c} a \cdot y_{1} & a \cdot y_{2} & a \cdot y_{3} & \dots & a \cdot y_{n} \end{array}] \end{aligned}

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Remark. The left argument must always be a scalar, and the right argument must always be a vector.
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