A scalar multiple is the product of a scalar (a real number) and a vector or matrix. It scales the magnitude of the vector or matrix without changing its direction.
congrats on reading the definition of scalar multiple. now let's actually learn it.
The scalar multiple of a vector $\mathbf{v}$ by a scalar $k$ is given by $k\mathbf{v} = (k v_1, k v_2, ..., k v_n)$ where $\mathbf{v} = (v_1, v_2, ..., v_n)$.
Scalar multiplication of a matrix $A$ by a scalar $c$ is performed element-wise: $cA = c[a_{ij}]$ resulting in $[ca_{ij}]$.
If the scalar is negative, it reverses the direction of the vector but maintains its proportionality.
In geometric applications, scaling vectors by different scalars can represent transformations such as stretching or compressing along certain directions.
Scalar multiples preserve linearity; that is, for vectors $\mathbf{u}$ and $\mathbf{v}$ and scalars $a$ and $b$, $(a+b)\mathbf{u} = a\mathbf{u} + b\mathbf{u}$.
Review Questions
What happens to a vector when it is multiplied by a negative scalar?
How do you compute the scalar multiple of a 3x3 matrix?
Can scalar multiplication change the direction of a vector? If so, how?
Related terms
Vector: An object with both magnitude and direction, often represented as an ordered list of numbers.
Matrix: A rectangular array of numbers arranged in rows and columns used to represent linear transformations and systems of equations.
Linear Transformation: A mapping between two modules (including vector spaces) that preserves addition and scalar multiplication.