Scalar Multiplication (Vector)

linear-algebra

Definition

Scalar Multiplication (Vector)

Let $V$ be a vector space over a field $F$. Scalar-vector multiplication is the operation

$$\cdot :F\times V\to V,$$

under which

• the map $v\mapsto \lambda v$ is linear for each $\lambda \in F$,
• the map $\lambda \mapsto \lambda v$ is additive, and
• $1_F\cdot v=v$.

Geometric Interpretation

Geometrically, scalar-vector multiplication scales a vector along its own line: $\lambda \mathbf{v}$ keeps the direction of $\mathbf{v}$ for $\lambda >0$, reverses it for $\lambda <0$, and collapses to the origin for $\lambda =0$. The magnitude scales by $|\lambda |$.

For $\lambda >0$, $\lambda \mathbf{v}$ lies on the ray through $\mathbf{v}$; for $\lambda <0$ it lies on the opposite ray; $\lambda =0$ collapses every vector to the origin.