Vectors

A vector is a special matrix with just one row or one column. Vectors are the building blocks of matrix operations.

Column Vectors

A column vector is a matrix with one column:

\[\begin{split} \mathbf{v} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \end{split}\]

This is a 3-element column vector. We often use bold lowercase letters for vectors.

Row Vectors

A row vector is a matrix with one row:

\[ \mathbf{u} = \begin{bmatrix} 4 & 5 & 6 \end{bmatrix} \]

Vectors in Python

Python Example

Create vectors and inspect their shape

Run

import numpy as np # Column vector (shape: 3x1) v = np.array([[1], [2], [3]]) # Or more commonly, as a 1D array v = np.array([1, 2, 3]) print("Vector v:", v) print("Shape:", v.shape)

Output Ready

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What Do Vectors Represent?

Vectors have many interpretations:

• Points in space: A 2D vector \([x, y]\) represents a point or position

• Directions: Vectors show direction and magnitude (like velocity or force)

• Data samples: A list of measurements (temperatures, prices, sensor readings)

• Features: In machine learning, a vector can represent features of an object

Engineering and Science Examples

Vectors can be used to represent quanties that posses both magnitude and direction. (It’s really hard not think about Despicable Me! whilst writing this).

Lets think of some real examples of quantities that could be reprented by vectors:

• Velocity: Speed in a specific direction

  • Wind speed & direction

  • Car speed & direction

• Acceleration: The rate and direction of a change in velocity

  • Rocket

• Forces

  • Gravity

  • Weight

  • Torque

• Electicity & Magnetism

  • Electic / Magnetic Field

The list goes on!!

Geometric Intuition

A 2D vector \(\mathbf{v} = [3, 2]\) can be visualized as an arrow from the origin to the point \((3, 2)\):

Python Example

Draw a 2D vector

Run

import matplotlib.pyplot as plt import numpy as np # Define a vector v = np.array([3, 2]) # Plot plt.figure(figsize=(6, 6)) plt.quiver(0, 0, v[0], v[1], angles='xy', scale_units='xy', scale=1, color='blue', width=0.006) plt.xlim(-1, 4) plt.ylim(-1, 3) plt.grid(True) plt.axhline(0, color='black', linewidth=0.5) plt.axvline(0, color='black', linewidth=0.5) plt.xlabel('x') plt.ylabel('y') plt.title('Vector v = [3, 2]') plt.show()

Output Ready

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This vector could represent the speed and direction of a moving vehicle or windspeed!!

Vector Operations

Just like numbers, we can add, subtract, and scale vectors:

Python Example

Add and scale vectors

Run

import numpy as np u = np.array([1, 2]) v = np.array([3, 1]) # Addition w = u + v print("u + v =", w) # [4, 3] # Scaling (multiply by a number) scaled = 2 * u print("2 * u =", scaled) # [2, 4]

Output Ready

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Why Vectors Matter

Vectors are everywhere in matrix mathematics:

• Matrices can be thought of as collections of column or row vectors

• Matrix multiplication combines vectors in powerful ways

• Many transformations act on vectors to produce new vectors

Understanding vectors is essential for understanding what matrices do.

Try It

In the vector plot, change v = np.array([3, 2]) to a few different vectors:

• [-2, 3]

• [4, -1]

• [0, 3]

Before running each one, predict the direction of the arrow.

Key Takeaways

• A vector is a one-dimensional list of values.

• Vectors can represent points, directions, measurements, or features.

• Adding vectors combines their components.

• Scaling a vector changes its length without changing its basic direction, unless the scale is negative.

Next Steps

Now let’s look at Matrix Shape and Size to understand how matrices are sized and structured.