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Vectors in Euclidean space

Vectors as objects with length and direction: adding and scaling them, normalising to unit vectors, the dot product and the norm it defines, orthogonality and independence, span and basis, and projection onto a line.

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The dot product

Learn to compute the dot product of two vectors from their components, read what it tells you about the angle between them, and divide out the lengths to find the cosine of that angle.

\vec{v} \cdot \vec{w} = |\vec{v}|\,|\vec{w}| \cos(\theta)

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Projecting vectors onto a line

Learn how to calculate the scalar and vector projection of one vector onto another in any dimension, interpret their geometric meaning, and understand their importance in measuring how much a vector aligns with a chosen direction, especially in machine learning contexts.

\text{proj}_{\vec{w}} \vec{v} = \dfrac{\vec{v} \cdot \vec{w}}{\|\vec{w}\|}

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Lesson 05 · The dot product

0,0

\vec{v}

\vec{w}

θ

Drag either arrowhead. Pulling it further out changes that length alone; swinging it round changes the angle alone.

‖ \vec{v} ‖ = 4.12

\times

‖ \vec{w} ‖ = 3.16

\times

\cos (58^{\circ}) = 0.537

=

\vec{v} \cdot \vec{w} = 7.00

oppositeright anglesame direction

An acute angle, so the dot product is positive

The components give the same number:

(4.0) (1.0) + (1.0) (3.0) = 7.00

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The 10 lessons

The 10 lessons in this unit. Expand any one to see what it covers. Each unlocks when its own prerequisites are passed.

Introduction to vectors

9 questions

Represent data as a column of numbers, read a $2$ -dimensional vector as an arrow on the coordinate plane, and say when two vectors are equal.

Covers

What is a vector?

Vectors on the coordinate plane

Vector length and unit vectors

8 questions

Compute the length of a vector with any number of components, and rescale a non-zero vector to a unit vector of length $1$ pointing the same way.

Covers

The length of a vector

Unit vectors and normalisation

Adding and scaling two-dimensional vectors

15 questions

Understand how to add, subtract and scale two-dimensional vectors componentwise and use these operations to form linear combinations, which are fundamental to representing and manipulating data in machine learning.

Covers

Adding and subtracting vectors

Scalar multiplication

Linear combinations of vectors

Adding and scaling $n$ -dimensional vectors

14 questions

Learn how to add, subtract and scale vectors of any dimension component by component, and how to evaluate a linear combination of several of them.

Covers

Adding and subtracting $n$ -dimensional vectors

Scalar multiplication

Linear combinations of vectors

The dot product

Preview

13 questions

Learn to compute the dot product of two vectors from their components, read what it tells you about the angle between them, and divide out the lengths to find the cosine of that angle.

Covers

The dot product

What the dot product says about the angle

The cosine of the angle

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Orthogonal, dependent and independent vectors

9 questions

Learn how to test whether pairs or sets of vectors are orthogonal (dot product zero) or linearly dependent (one is a scalar multiple or linear combination of others), and understand the implications of these properties for redundancy and uniqueness in feature spaces such as those used in machine learning.

Covers

Orthogonality in $2$ dimensions and $n$ dimensions

Linear dependence / independence of two vectors

Linear dependence / independence of a set of vectors

Span of vectors in $n-$ dimensional space

9 questions

Understand how the span of a set of vectors describes all the points reachable by their linear combinations, how to determine if a given vector lies within this span, and how to check if a set of vectors spans the entire space $\mathbb{R}^N$ using row reduction.

Covers

Definition of span and geometric intuition

Is a vector in the span?

Does a set of vectors span all of $\mathbb{R}^n$ ?

Projecting vectors onto a line

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10 questions

Covers

Geometric meaning of projection in $2$ dimensions

Scalar and vector projection in $2$ dimensions

Generalisation to $n-$ dimensional vectors

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Bases and orthonormal bases

6 questions

Understand how bases provide reference directions for vector spaces and see how orthonormal bases simplify the representation of vectors and calculations such as finding coordinates using dot products.

Covers

What is a basis?

Orthonormal basis

Changing basis

6 questions

Learn how to represent a vector in different bases by finding its coordinates relative to any basis and use the basis matrix and its inverse to convert between these coordinate systems and the standard basis.

Covers

Coordinates in a new basis

Matrix form of change of basis

Where this unit sits in the curriculum

This unit is part of our Essential Linear Algebra for ML learning path, which contains 65 lessons. Every one of them is drawn below.

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10 lessons in this unit

34 prerequisite lessons across the unit, counting every step back to the start

21 other lessons in the learning path - after this unit, or alongside it

Every dot is a lesson, every line a prerequisite. You can only start a lesson once you have mastered all of its prerequisites, so you are always building on solid foundations.

Builds on: Exponents and roots, Graphs of functions, Matrices and linear systems, Matrix transformations, Trigonometry

Leads to: Eigenvalues, eigenvectors and singular value decomposition, Matrices, Optimisation Foundations

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