Column Vectors: The Building Blocks of Linear Algebra
What Exactly is a Column Vector?
Let's start with the simplest definition from our topic: A column vector is a matrix with a single column. But what is a matrix? Think of a matrix as a rectangular grid of numbers, like a spreadsheet or a chessboard filled with numbers. A column vector is just a special case of that grid where there's only one column.
For example, here is a column vector with 3 rows:
This vector has three entries: 2, 5, and -1, stacked vertically. We say its dimensions are 3 × 1 (read as "three by one"), meaning 3 rows and 1 column.
We use square brackets $[ ]$ or parentheses $( )$ to enclose the numbers. The numbers inside are called elements or components. Each element's position is important. The top element is the first component, the one below it is the second, and so on.
We often name vectors with bold lowercase letters, like v, or with an arrow, like $\vec{v}$. For example:
$\vec{v} = \begin{bmatrix} 2 \\ 5 \\ -1 \end{bmatrix}$ or x $= \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$, where $x_1, x_2, x_3$ can be any numbers.
Column Vectors vs. Row Vectors
If a column vector is a matrix with one column, what do you think a row vector is? That's right—it's a matrix with a single row! The orientation matters. A column vector stands up, and a row vector lies down.
| Type | What it is | Dimensions | Example |
|---|---|---|---|
| Column Vector | A matrix with one column. | $m \times 1$ (e.g., 3 × 1) | $\begin{bmatrix} 7 \\ 4 \end{bmatrix}$ |
| Row Vector | A matrix with one row. | $1 \times n$ (e.g., 1 × 4) | $\begin{bmatrix} 5 & -2 & 0 & 9 \end{bmatrix}$ |
| Relation | A row vector is the transpose of a column vector, and vice versa. The transpose operation, denoted by a superscript $T$, flips the rows and columns. If $\vec{v} = \begin{bmatrix} 7 \\ 4 \end{bmatrix}$, then $\vec{v}^T = \begin{bmatrix} 7 & 4 \end{bmatrix}$. | ||
Basic Operations with Column Vectors
You can perform cool math operations with column vectors, similar to how you work with regular numbers, but with some special rules.
1. Addition and Subtraction: You can only add or subtract column vectors that have the same dimensions. You simply add or subtract the corresponding elements.
Example: Let $\vec{a} = \begin{bmatrix} 3 \\ 0 \\ 5 \end{bmatrix}$ and $\vec{b} = \begin{bmatrix} -1 \\ 4 \\ 2 \end{bmatrix}$.
Then, $\vec{a} + \vec{b} = \begin{bmatrix} 3 + (-1) \\ 0 + 4 \\ 5 + 2 \end{bmatrix} = \begin{bmatrix} 2 \\ 4 \\ 7 \end{bmatrix}$.
2. Scalar Multiplication: You can multiply a column vector by a single number (called a scalar). Multiply every element in the vector by that scalar.
Example: If $k = 3$ and $\vec{v} = \begin{bmatrix} 1 \\ -2 \end{bmatrix}$, then $k\vec{v} = 3 \times \begin{bmatrix} 1 \\ -2 \end{bmatrix} = \begin{bmatrix} 3 \times 1 \\ 3 \times (-2) \end{bmatrix} = \begin{bmatrix} 3 \\ -6 \end{bmatrix}$.
3. Dot Product (Inner Product): This is a way to multiply a row vector and a column vector of the same length to get a single number. It's incredibly important in physics and geometry.
If $\vec{u} = \begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bmatrix}$ and $\vec{v} = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix}$, their dot product is: $\vec{u} \cdot \vec{v} = u_1 v_1 + u_2 v_2 + u_3 v_3$.
Important: Technically, for the dot product, we treat the first vector as a row (transpose it): $\vec{u}^T \vec{v}$.
Example: $\begin{bmatrix} 2 \\ -1 \\ 4 \end{bmatrix} \cdot \begin{bmatrix} 3 \\ 0 \\ 5 \end{bmatrix} = (2\times3) + (-1\times0) + (4\times5) = 6 + 0 + 20 = 26$.
Seeing Vectors in the World Around Us
Column vectors are not just abstract math symbols. They are used everywhere to represent real-world things that have both a magnitude (size) and a direction, or simply to list related data.
1. Physics - Force and Movement: In physics, a force is a perfect example of a vector. A force of 10 N pushing to the right can be represented in 2D as $\vec{F} = \begin{bmatrix} 10 \\ 0 \end{bmatrix}$, where the first component is the horizontal part and the second is the vertical part. Velocity and acceleration are also vectors.
2. Computer Graphics - Points in Space: Every pixel or point in a 2D video game or a 3D animated movie is stored as a vector! A point in 3D space has coordinates $(x, y, z)$, which are neatly stored as a column vector: $\begin{bmatrix} x \\ y \\ z \end{bmatrix}$. To move (translate) or rotate an object, we perform mathematical operations on all the point vectors that make up that object.
3. Data Science - Feature Vectors: Imagine you are collecting data about houses to predict their price. For each house, you might record its size (in m²), number of bedrooms, and age (in years). One house could be represented as a column vector: $\text{house} = \begin{bmatrix} 120 \\ 3 \\ 10 \end{bmatrix}$. In machine learning, each piece of data (a house, a photo, a customer) is often a column vector, making it easy for computers to process.
4. Simple Everyday Lists: Even a grocery list with quantities can be thought of as a vector! If you need 2 apples, 1 bread, and 3 milks, your "shopping vector" is $\begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix}$.
A Practical Example: Navigating a City Grid
Let's use column vectors to solve a fun problem. Imagine you are on a giant grid, like the streets of New York City. Your position can be given by $(x, y)$ coordinates.
You start at point A. Your friend gives you directions as column vectors, where the top number is "blocks East" (positive) or West (negative), and the bottom number is "blocks North" (positive) or South (negative).
Let your starting position be $\vec{A} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ (the origin).
First, you walk 3 blocks East and 1 block North. This movement is $\vec{m}_1 = \begin{bmatrix} 3 \\ 1 \end{bmatrix}$.
Your new position is $\vec{A} + \vec{m}_1 = \begin{bmatrix} 0+3 \\ 0+1 \end{bmatrix} = \begin{bmatrix} 3 \\ 1 \end{bmatrix}$.
Then, you walk 1 block West and 4 blocks North. This is $\vec{m}_2 = \begin{bmatrix} -1 \\ 4 \end{bmatrix}$.
Your final position $\vec{F}$ is: $\vec{F} = \begin{bmatrix} 3 \\ 1 \end{bmatrix} + \begin{bmatrix} -1 \\ 4 \end{bmatrix} = \begin{bmatrix} 2 \\ 5 \end{bmatrix}$.
So you are now at coordinates $(2, 5)$ on the grid. We used vector addition to combine the two movement steps into one net result!
What is the total distance "as the crow flies" from start to finish? That's the magnitude (or length) of the total displacement vector $\vec{d} = \begin{bmatrix} 2 \\ 5 \end{bmatrix}$. We find it using the Pythagorean theorem: $|\vec{d}| = \sqrt{2^2 + 5^2} = \sqrt{4 + 25} = \sqrt{29} \approx 5.39$ blocks.
Important Questions
Yes, absolutely! A column vector with just one number is a 1 × 1 matrix. It's like a single point on a number line. For example, $\begin{bmatrix} 7 \end{bmatrix}$ is a perfectly valid column (and also row) vector. It represents a one-dimensional quantity.
The vertical column format is crucial for two main reasons. First, it aligns with how we set up systems of linear equations. The coefficients in front of a single variable are naturally stacked. Second, in computing and advanced math, when we multiply a larger matrix by a column vector, the rule "rows of the matrix dot product with the column vector" works perfectly with this vertical structure. It provides a consistent and efficient way to organize data for calculation.
A list is just a collection. A vector is a list plus a set of rules for how to interact with other vectors (like addition and scalar multiplication). The mathematical structure of a vector space[1] gives vectors their power. For instance, you can't naturally "add" two grocery lists together in a meaningful way, but you can add two force vectors to find the net force.
Footnote
[1] Vector Space: A collection of objects (vectors) that can be added together and multiplied by scalars (numbers), and these operations follow specific rules like commutativity and associativity. It is the formal mathematical framework that makes linear algebra work.
[2] Transpose (T): An operation that flips a matrix over its main diagonal, turning rows into columns and columns into rows. For a column vector, the transpose is a row vector.
[3] Scalar: In the context of vectors, a scalar is a single real number (not a vector) used to scale (multiply) a vector, changing its magnitude but not its direction (unless the scalar is negative, which reverses direction).