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Multiplying a Vector by a Matrix

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Multiplying a Vector by a Matrix

Study Guide

Key Definition

In mathematics, a vector is a matrix with one row or one column. We treat vectors as column vectors by default. To compute A·v, the number of columns of A must equal the number of rows of v.

Important Notes

A $vector$ is a list of numbers arranged in a row or column.
A $matrix$ can contain multiple vectors.
For multiplication, the number of columns in $\mathbf{A}$ must equal the number of rows in $\mathbf{v}$.
Geometrically, multiplying by $\mathbf{A}$ maps vectors by scaling, rotating, or shearing in space.
If the result is the zero vector, then $\mathbf{v}$ lies in the null space of $\mathbf{A}$.

Mathematical Notation

$\times$ represents multiplication

$\cdot$ represents dot product

$\mathbf{v}$ represents a vector

$\mathbf{A}$ represents a matrix

$\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is a 2x2 matrix

Remember to use proper notation when solving problems

Why It Works

Multiplying a vector by a matrix applies the linear transformation defined by the matrix to the vector, combining its rows with the vector components.

Remember

Every vector can be considered a special case of a matrix with one column, enabling the use of matrix operations.

Quick Reference

Matrix Multiplication:$\mathbf{A} \cdot \mathbf{v}$

Understanding Multiplying a Vector by a Matrix

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Multiplying a 2×2 matrix A by a 2×1 column vector v produces a 2×1 vector. Check that A has 2 columns and v has 2 rows. For example, $A=\begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}$, $v=\begin{bmatrix}5\\6\end{bmatrix}$. Compute entry 1: 1×5+2×6=17; entry 2: 3×5+4×6=39. Result: $\begin{bmatrix}17\\39\end{bmatrix}$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the result of $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \cdot \begin{bmatrix} 5 \\ 6 \end{bmatrix}$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine a robot arm being controlled by vectors and matrices to move accurately. How does $\begin{bmatrix} 2 & 3 \\ 1 & 0 \end{bmatrix} \cdot \begin{bmatrix} x \\ y \end{bmatrix}$ change the movement?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Challenge: If $\mathbf{A}$ is a 3x3 matrix and $\mathbf{v}$ is a 3x1 vector, what must be true for $\mathbf{A} \cdot \mathbf{v} = \mathbf{0}$?

Challenge Quiz

Single Choice Quiz

Advanced

Given $\begin{bmatrix} a & b \\ c & d \end{bmatrix} \cdot \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$, what can be inferred?

Recap

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