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Vectors in Euclidean space

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

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Dot product and vector norm

Learn how to compute the dot product of vectors using both magnitude and angle or components, understand its geometric meaning in terms of similarity and orientation, and use the dot product to define vector norms and create unit vectors for comparing directions irrespective of scale.

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

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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.

$\text{proj}_{\vec{w}} \vec{v} = |\vec{v}| \cos(\theta)$

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Lesson 04 · Dot product and vector norm

0,0

\vec{v} = [\begin{matrix} 2.00 \\ 1.00 \end{matrix}]

\vec{w} = [\begin{matrix} 2.00 \\ - 1.00 \end{matrix}]

θ = {53.1}^{\circ}

| \vec{v} | = 2.24, | \vec{w} | = 2.24

\vec{v} \cdot \vec{w} = 3.00

The arc shows the smaller angle $θ$ between $\vec{v}$ and $\vec{w}$ . This is the angle used in the dot product formula: $\vec{v} \cdot \vec{w} = | \vec{v} | | \vec{w} | \cos θ$

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The 9 lessons

The 9 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

Understand how data can be represented as vectors, how to visualise and interpret vectors in two dimensions, calculate their magnitude, and convert them into unit vectors through normalisation.

Covers

Visualizing $2$ -dimensional vectors

The length of a vector

Unit vectors and normalisation

Addition, subtraction and scalar multiplication of $2-$ dimensional vectors

12 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

Vector addition

Vector subtraction

Scalar multiplication

Linear combinations of vectors

Addition, subtraction and scalar multiplication of $N-$ dimensional vectors

12 questions

Learn how to add, subtract and scale $n$ -dimensional real vectors entrywise, and construct linear combinations by applying these operations in any number of dimensions.

Covers

$n$ -dimensional real vectors

Addition and subtraction of $n$ -dimensional real vectors

Scalar multiplication of $n$ -dimensional real vectors

Linear combinations of $n$ -dimensional real vectors

Dot product and vector norm

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9 questions

Covers

Dot product using magnitude and angle

Dot product using components

Vector norms and unit vectors

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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 61 lessons. Every one of them is drawn below.

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9 lessons in this unit

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

19 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: Matrices and linear systems, Matrix transformations, Set operations, Trigonometry, What is Machine Learning?

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

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More in Linear algebra: Matrices, Matrices and linear systems, Matrix transformations, Eigenvalues, eigenvectors and singular value decomposition