Scalar Multiplication

Multiply a vector by a scalar (number):

$$c\mathbf{v} = \begin{pmatrix} cv_1 \\ cv_2 \end{pmatrix}$$

Effect

• $c > 1$: stretches the vector
• $0 < c < 1$: shrinks the vector
• $c < 0$: reverses direction and scales
• $c = -1$: flips the vector

Properties

• Distributive: $c(\mathbf{u} + \mathbf{v}) = c\mathbf{u} + c\mathbf{v}$
• Distributive: $(c + d)\mathbf{v} = c\mathbf{v} + d\mathbf{v}$
• Associative: $c(d\mathbf{v}) = (cd)\mathbf{v}$