Vector and Matrix Operations

Behind every operation rule lies meaning: addition represents composition, scalar multiplication represents scaling, the dot product measures similarity, and matrix multiplication composes relationships.

I always verify dimensions first—and then interpret the meaning of the result. Just because an operation is computable doesn’t mean its interpretation is correct.

In the previous article, we explored the definitions and representations of vectors and matrices. This article focuses on fundamental operations involving vectors and matrices—helping you better understand how to apply these mathematical tools to real-world problems.

Review of Basic Concepts: Vectors and Matrices

Before diving into operations, let’s briefly revisit the core concepts.

• Vector: A vector is an ordered collection of numbers, typically represented as an $n \times 1$ column matrix. For example, $\mathbf{v} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$ is a three-dimensional vector.

• Matrix: A matrix is a two-dimensional array of numerical values. An $m \times n$ matrix is written as

  $$A = \begin{bmatrix} a_{11} & a_{12} & \ldots & a_{1n} \\ a_{21} & a_{22} & \ldots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \ldots & a_{mn} \end{bmatrix}.$$

Vector Operations

Vectors support several key operations—the most common being vector addition, scalar multiplication, and the dot (inner) product.

Vector Addition

Given two vectors $\mathbf{u} = \begin{bmatrix} u_1 \\ u_2 \\ \vdots \\ u_n \end{bmatrix}$ and $\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}$, their sum $\mathbf{u} + \mathbf{v}$ yields a new vector $\mathbf{w}$:

$$\mathbf{w} = \mathbf{u} + \mathbf{v} = \begin{bmatrix} u_1 + v_1 \\ u_2 + v_2 \\ \vdots \\ u_n + v_n \end{bmatrix}$$

Example

Let $\mathbf{u} = \begin{bmatrix} 1 \\ 3 \end{bmatrix}$ and $\mathbf{v} = \begin{bmatrix} 4 \\ 2 \end{bmatrix}$:

$$\mathbf{w} = \mathbf{u} + \mathbf{v} = \begin{bmatrix} 1 + 4 \\ 3 + 2 \end{bmatrix} = \begin{bmatrix} 5 \\ 5 \end{bmatrix}$$

Scalar Multiplication of Vectors

Given a scalar $k$ and a vector $\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}$, scalar multiplication yields:

$$k \mathbf{v} = \begin{bmatrix} k v_1 \\ k v_2 \\ \vdots \\ k v_n \end{bmatrix}$$

Example

Let $k = 3$ and $\mathbf{v} = \begin{bmatrix} 2 \\ 4 \end{bmatrix}$:

$$k \mathbf{v} = 3 \begin{bmatrix} 2 \\ 4 \end{bmatrix} = \begin{bmatrix} 3 \cdot 2 \\ 3 \cdot 4 \end{bmatrix} = \begin{bmatrix} 6 \\ 12 \end{bmatrix}$$

Dot Product (Inner Product) of Vectors

The dot product is a fundamental operation. For vectors $\mathbf{u}$ and $\mathbf{v}$, it is defined as:

$$\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + \ldots + u_n v_n$$

Example

Let $\mathbf{u} = \begin{bmatrix} 1 \\ 3 \\ -5 \end{bmatrix}$ and $\mathbf{v} = \begin{bmatrix} 4 \\ -2 \\ -1 \end{bmatrix}$:

$$\mathbf{u} \cdot \mathbf{v} = 1 \cdot 4 + 3 \cdot (-2) + (-5) \cdot (-1) = 4 - 6 + 5 = 3$$

Matrix Operations

Matrix operations include matrix addition, scalar multiplication, and matrix multiplication. Below, we cover addition and scalar multiplication in detail.

When performing vector or matrix operations, always first confirm: dimensions, orientation, output shape, and contextual meaning. Dimensional compatibility is only the first step—semantic alignment is the key.

Matrix Addition

For two matrices $A$ and $B$ of identical dimensions, their sum $C = A + B$ is computed element-wise:

$$C = A + B = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} & \ldots \\ a_{21} + b_{21} & a_{22} + b_{22} & \ldots \\ \vdots & \vdots & \ddots \end{bmatrix}$$

Example

Let $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$. Then:

$$C = A + B = \begin{bmatrix} 1 + 5 & 2 + 6 \\ 3 + 7 & 4 + 8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}$$

Scalar Multiplication of Matrices

Given a scalar $k$ and a matrix $A = \begin{bmatrix} a_{ij} \end{bmatrix}$, scalar multiplication is defined as:

Before reading “Vector and Matrix Operations”, preview the visual path from problem → computation → result shown in this diagram. After reading, revisit the text to verify whether you can reproduce each step.

$$kA = \begin{bmatrix} k a_{11} & k a_{12} & \ldots \\ k a_{21} & k a_{22} & \ldots \\ \vdots & \vdots & \ddots \end{bmatrix}$$

Example

Let $k = 2$ and $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$. Then:

$$kA = 2A = \begin{bmatrix} 2 \cdot 1 & 2 \cdot 2 \\ 2 \cdot 3 & 2 \cdot 4 \end{bmatrix} = \begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}$$

Matrix Multiplication

Matrix multiplication combines rows of the first matrix with columns of the second. If $A$ is $m \times n$ and $B$ is $n \times p$, then the product $AB$ is an $m \times p$ matrix.

Example

Let

$$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 0 \\ 1 & 2 \end{bmatrix}.$$

Then:

$$AB = \begin{bmatrix} 1 \cdot 2 + 2 \cdot 1 & 1 \cdot 0 + 2 \cdot 2 \\ 3 \cdot 2 + 4 \cdot 1 & 3 \cdot 0 + 4 \cdot 2 \end{bmatrix} = \begin{bmatrix} 4 & 4 \\ 10 & 8 \end{bmatrix}$$

When reviewing “Vector and Matrix Operations”, place key concepts, procedural steps, and observable outcomes on the same page for efficient recall.

When practicing “Vector and Matrix Operations”, write down input conditions, computational actions, and resulting outputs together—making future review and verification straightforward.

Summary

Vector and matrix operations are not isolated formulas. Vector addition and scalar multiplication help us reason about direction and scaling; the dot product quantifies similarity; and matrix multiplication composes multiple linear relationships. In practice, always begin by verifying dimensional compatibility, and then interpret the business or geometric meaning of the computed result.