Multivariable calculus and vector analysis

Dot product in matrix notation

The transpose of a matrix

Before describing the dot product in matrix notation, we briefly digress to make sure you know what the transpose of a matrix is.

We can transpose a matrix by switching its rows with its columns. We denote the transpose of matrix $A$ by $A^T$. For example, if

$$\begin{array}{lll}\hfill A=\left[\begin{array}{ccc}\hfill 1& \hfill 2& \hfill 3\\ \hfill 4& \hfill 5& \hfill 6\end{array}\right]& & \hfill \end{array}$$

then the transpose of $A$ is

$$\begin{array}{lll}\hfill A^T=\left[\begin{array}{cc}\hfill 1& \hfill 4\\ \hfill 2& \hfill 5\\ \hfill 3& \hfill 6\end{array}\right].& & \hfill \end{array}$$

Since an $n$-dimensional vector $\mathbf{x}$ is represented by an $n\times 1$ column matrix,

$$\begin{array}{lll}\hfill \mathbf{x}=\left[\begin{array}{c}\hfill x_1\hfill \\ \hfill x_2\hfill \\ \hfill x_3\hfill \\ \hfill ⋮\hfill \\ \hfill x_n\hfill \end{array}\right],& & \hfill \end{array}$$

the transpose ${\mathbf{x}}^T$ is a $1\times n$ row matrix

$$\begin{array}{lll}\hfill {\mathbf{x}}^T=\left[\begin{array}{ccccc}\hfill x_1\hfill & \hfill x_2\hfill & \hfill x_3\hfill & \hfill \cdots \hfill & \hfill x_n\hfill \end{array}\right].& & \hfill \end{array}$$

The dot product as matrix multiplication

Given the rules of matrix multiplication, the product of a $1\times n$ matrix with an $n\times 1$ matrix is a $1\times 1$ matrix, i.e., a scalar. So if we multiply ${\mathbf{x}}^T$ (a $1\times n$ matrix) with any $n$-dimensional vector $\mathbf{y}$ (viewed as an $n\times 1$ matrix), we end up with a matrix multiplication equivalent to the familiar dot product of $\mathbf{x}\cdot \mathbf{y}$:

$$\begin{array}{lll}\hfill {\mathbf{x}}^T\mathbf{y}=\left[\begin{array}{ccccc}\hfill x_1\hfill & \hfill x_2\hfill & \hfill x_3\hfill & \hfill \cdots \hfill & \hfill x_n\hfill \end{array}\right]\left[\begin{array}{c}\hfill y_1\hfill \\ \hfill y_2\hfill \\ \hfill y_3\hfill \\ \hfill ⋮\hfill \\ \hfill y_n\hfill \end{array}\right]=x_1{y}_1+x_2{y}_2+x_3{y}_3+\dots +x_n{y}_n=\mathbf{x}\cdot \mathbf{y}.& & \hfill \end{array}$$

Although we won’t typically write a dot product as ${\mathbf{x}}^T\mathbf{y}$, you may see it elsewhere. Moreover, this dot product forms the building block for the general matrix multiplication.

Contents by topic

1. Geometry in higher dimensions

1a. Preliminaries

1b. Lines and planes

1c. Matrices

• Matrices and determinants
• Multiplying matrices and vectors
• Dot product in matrix notation
• Matrices and linear functions
• The relationship between determinants and area or volume
• Matrix and vector multiplication examples

1d. Visualizing functions in higher dimensions

2. Differential Calculus

3. Integral calculus

4. Vector calculus

5. The fundamental theorems of vector calculus

6. Review material