Scalar multiplication

Definition

Scalar multiplication involves multiplying a vector by a scalar (a single number), resulting in a new vector where each component is scaled by the scalar. This operation changes the magnitude of the vector but not its direction if the scalar is positive.

5 Must Know Facts For Your Next Test

1. Multiplying a vector $\mathbf{v}$ by a scalar $k$ creates a new vector $k\mathbf{v}$ where each component of $\mathbf{v}$ is multiplied by $k$.
2. If the scalar is negative, it reverses the direction of the vector.
3. The magnitude of the resulting vector $k\mathbf{v}$ is $|k|$ times the magnitude of $\mathbf{v}$.
4. Scalar multiplication can be represented as stretching or shrinking a vector depending on whether $|k| > 1$ or $0 < |k| < 1$, respectively.
5. In matrix operations, scalar multiplication applies to every element within the matrix.

Review Questions

• What happens to a vector when it is multiplied by a negative scalar?
• How does scalar multiplication affect the magnitude and direction of a vector?
• Describe how you would perform scalar multiplication on a matrix.

Related terms

Vector: A quantity that has both magnitude and direction, often represented as an arrow in space or as an ordered list of components.

Matrix Multiplication: An operation that takes two matrices and produces another matrix by taking dot products of rows and columns.

\textbackslash textit{Dot Product}: \textbackslash textit{An algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number.}