Scalar Multiplication¶

Definition¶

Scalar multiplication is the operation that takes a scalar $\alpha\in\mathbb{F}$ and a vector $\boldsymbol{v}\in V$ and returns a vector $\alpha \boldsymbol{v}\in V$ $V$ .

Interpretation¶

Scalar multiplication rescales a vector by a number from the field: in a real coordinate space it stretches or shrinks the vector, a negative scalar also reverses its direction, and in coordinates every entry is multiplied by the same number, $\alpha(v_1,\ldots,v_n)=(\alpha v_1,\ldots,\alpha v_n)$ $(α v_{1}, \dots, α v_{n})$ .

A vector v = (2, 1) on a grid with its multiples 2v, half of v and -v, all lying on the same line through the origin.

The figure shows $\boldsymbol{v}=(2,1)$ with $2\boldsymbol{v}=(4,2)$ $(4, 2)$ , twice as long, $\tfrac{1}{2}\boldsymbol{v}=(1,\tfrac{1}{2})$ $(1, \frac{1}{2})$ , half as long, and $-\boldsymbol{v}=(-2,-1)$ $(- 2, - 1)$ , the same length in the opposite direction. All multiples $\alpha\boldsymbol{v}$ lie on the dashed line through the origin and $\boldsymbol{v}$ , and as $\alpha$ runs over $\mathbb{R}$ they fill exactly this line.

The vector space axioms $1\boldsymbol{v}=\boldsymbol{v}$ $v$ , $(\alpha\beta)\boldsymbol{v}=\alpha(\beta\boldsymbol{v})$ and the two distributive laws tie it to the other operations; from them follow $0\boldsymbol{v}=\boldsymbol{0}$ and $(-1)\boldsymbol{v}=-\boldsymbol{v}$ $- v$ . In a general vector space it acts on whatever the vectors are: for polynomials, $3(1+x)=3+3x$ $3 x$ . It is not a product of two vectors: the vector space axioms provide no such product, even though some spaces, such as polynomials, carry one as extra structure.

Key Results¶

$1\boldsymbol{v}=\boldsymbol{v}$ $v$ .
$(\alpha\beta)\boldsymbol{v}=\alpha(\beta \boldsymbol{v})$ $α (β v)$ .

Examples¶

In $\mathbb{R}^3$ , $2(1,-1,4)=(2,-2,8)$ $(2, - 2, 8)$ .
For $\boldsymbol{v}=(2,-3)$ $(2, - 3)$ , $-\tfrac{1}{2}\boldsymbol{v}=(-1,\tfrac{3}{2})$ reverses the direction and halves the length, while $0\boldsymbol{v}=(0,0)$ gives the zero vector.

Prerequisites¶

Zero Vector: multiplying any vector by the zero scalar gives the zero vector.
Matrix Scalar Multiplication
Vector

Sources¶

Math II: Linear Algebra · Week 2

Aliases: multiplication by a scalar Symbols: alpha, beta, element of, real numbers

Last updated: 27 October 2026 11:52