# Linear Algebra - Scalar (Multiplication|Product) - Scaling

Scalar Multiplication (Scaling) is the multiplication of a vector (for instance $v_a$) by a scalar (real number) (for instance $\alpha$) to produce another vector (for instance $v_b$)

$f (v_a) = \alpha.v_a = v_b$

Multiplying a vector v by a scalar $\alpha$ is defined as multiplying each entry of v by $\alpha$ :

$\alpha.v = \alpha[v_{[1]},v_{[2]},\dots,v_{[n]}] = [\alpha.v_{[1]},\alpha.v_{[2]}, \dots ,\alpha.v_{[n]}]$

$v. \alpha$ is not legal whereas $\alpha . v$ is.

Scalar Multiplication is used to define a line

## 3 - Property

Scalar multiplication is:

• Associative: $\alpha(\beta.v) = (\alpha.\beta)v$

## 4 - Geometric Representation

The green arrow represents the vector [4, 1.5] and the red arrow represents two times this vector.

• Scalars bigger than 1 give rise to somewhat larger copies of the original vector
• Scalars smaller than 1 give rise to somewhat smaller copies of the original vector
• Negative scalars give rise to vectors pointing in the opposite direction

## 5 - Computation

2 [5, 4, 10] = [2 x 5, 2 x 4, 2 x 10] = [10, 8, 20]

Python:

def scalarVectorMult(alpha, v): return [alpha*x for x in v]