Linear Algebra
2.47 min read

Scalar Multiplication

Scalar multiplication scales a vector by a number (the "scalar"). To multiply v\mathbf{v} by cc, multiply every entry by cc: cv=(cv1,cv2,…,cvm)c\mathbf{v} = (cv_1, cv_2, \ldots, cv_m).

Geometrically, scalar multiplication scales the arrow: c>1c > 1 makes it longer, 0<c<10 < c < 1 shrinks it, c=−1c = -1 reverses direction, c=0c = 0 gives the zero vector.

Vectors of the form cvc\mathbf{v} for varying c∈Rc \in \mathbb{R} form a line through the origin — the set of all scalar multiples of v\mathbf{v}. This is the span of a single vector.

Formal View

Definition 2.4 — Scalar Multiplication
For c∈Rc \in \mathbb{R} and v∈Rm\mathbf{v} \in \mathbb{R}^m:
cv=(cv1⋮cvm)c\mathbf{v} = \begin{pmatrix} cv_1 \\ \vdots \\ cv_m \end{pmatrix}
The additive inverse is −v=(−1)v-\mathbf{v} = (-1)\mathbf{v}, with v+(−v)=0\mathbf{v} + (-\mathbf{v}) = \mathbf{0}.

Why This Matters

Scalar multiplication is how we encode "direction with magnitude" — the essence of quantities like velocity.

  • Scaling a force vector: twice the force in the same direction
  • Unit vectors: divide by magnitude to get a direction-only vector
  • Linear interpolation: (1−t)a+tb(1-t)\mathbf{a} + t\mathbf{b} is a path from a\mathbf{a} to b\mathbf{b}

Quiz

Question 1

What is −3⋅(2,−1,4)-3 \cdot (2, -1, 4)?

Common Mistakes

  • Applying scalar multiplication only to some entries — it must be applied to every entry uniformly.
  • Confusing c=0c = 0 (gives zero vector) with "the equation has a trivial solution."