2.68 min read

Applications: Physics, Graphics, Audio, Images

Vectors appear in every quantitative field. The power of the vector concept is that the same mathematics — addition, scalar multiplication, span — applies regardless of what the entries represent.

Physics: Velocity, force, momentum, and electric fields are all vectors in $\mathbb{R}^3$ . Newton's second law $F = ma$ becomes $\mathbf{F} = m\mathbf{a}$ in vector form.

Graphics: A 3D point has coordinates $(x, y, z)$ , transformations are matrix operations on these vectors. Colors are vectors $(r, g, b)$ in $[0,1]^3$ .

Audio: A digital audio clip at 44,100 samples/second for 1 second is a vector in $\mathbb{R}^{44100}$ . Filters are linear operations on this vector.

Images: A grayscale image of height $h$ and width $w$ is a vector in $\mathbb{R}^{hw}$ . Convolution filters are linear operations.

Formal View

Example 2.6 — Image as a Vector

100 \times 100

grayscale image has

10{,}000

pixels. It can be stored as a vector

\mathbf{x} \in \mathbb{R}^{10000}

where

x_i

is the brightness of pixel

i

. Blurring corresponds to computing

A\mathbf{x}

for an appropriate matrix

A

Why This Matters

The unifying power of vector notation is that one set of mathematical tools solves problems across all these domains.

Image compression (JPEG) uses linear algebra on image vectors
Audio equalization applies linear filters (matrix operations) to sound vectors
Computer vision represents images as vectors and applies linear transformations to classify them

Learning Resources

Linear transformations and matrices

3Blue1Brown

How linear transformations act on vectors in applications.

10 min

Quiz

Question 1

A $50 \times 50$ grayscale image stored as a vector lives in $\mathbb{R}^n$ where $n$ is:

Common Mistakes

Thinking vectors must be 2D or 3D — high-dimensional vectors are equally fundamental.
Treating images/audio as "special" objects rather than vectors subject to linear algebra.