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Linear Algebra: Vectors, Matrices and Transformations – AI Tutor-Enabled Course Book

Original price was: $54.00.Current price is: $35.00.

AI Tutor-enabled
Self-paced
Beginner → Advanced

Modules: 10
Study time: About 40 hours
Format: Digital course book (PDF)
Projects: 4 + capstone

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See the geometry behind the matrices, and master the methods used in data science and engineering.

TypeNotes & books

DeliveryInstant download after payment

Instant download after payment
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SKU: CB-LINALG Categories: Course Books, Mathematics Tags: eigenvalues, linear algebra course, linear algebra for data science, linear algebra study guide, matrices, vectors

AI Tutor-Enabled Self-Paced Professional Course Book

See the geometry behind the matrices, and master the methods used in data science and engineering.

Linear Algebra is a self-paced course book covering vectors, systems of equations, matrices, determinants, vector spaces, eigenvalues and orthogonality. Each idea is explained both algebraically and geometrically, with applications to data science, engineering and economics.

Is this you?

Row reduction feels like endless arithmetic
Vector spaces and span seem abstract
You can compute eigenvalues but not explain them
Proofs in linear algebra are unfamiliar
You need linear algebra for machine learning
Exams mix computation and theory

This course book is built to close exactly that gap: clear explanations, worked practice and an AI tutor that guides you instead of just giving answers.

Learn
Practise
Check with your AI Tutor
Build projects
Get assessed

What you’ll learn in 10 practical modules

01 VectorsWork with vectors in two, three and n dimensions.

Key concepts

Vector operations
Dot product and length
Lines and planes

Sample practice

Find the angle between (1,2) and (3,1)
Find the plane through three points

02 Systems of Linear EquationsSolve systems systematically.

Key concepts

Gaussian elimination
Row echelon forms
Consistent and inconsistent systems

Sample practice

Solve a 3×3 system by row reduction
Describe all solutions of an underdetermined system

03 Matrix AlgebraCompute with matrices.

Key concepts

Matrix operations
Inverses
Elementary matrices

Sample practice

Find A⁻¹ for a 2×2 matrix
Explain why AB ≠ BA in general

04 DeterminantsMeasure how a matrix scales space.

Key concepts

Cofactor expansion
Properties of determinants
Cramer’s rule

Sample practice

Compute a 3×3 determinant
Use det to decide if A is invertible

05 Vector Spaces and SubspacesUnderstand the structure behind solutions.

Key concepts

Span and linear independence
Basis and dimension
Column space and null space

Sample practice

Is {(1,2,3),(2,4,6)} independent?
Find a basis for a null space

06 Linear TransformationsTreat matrices as functions.

Key concepts

Kernel and range
Matrix of a transformation
Rotations and projections

Sample practice

Find the matrix of a 90° rotation
Find the kernel of a transformation

07 Eigenvalues and EigenvectorsFind the directions a matrix preserves.

Key concepts

Characteristic polynomial
Eigenvectors
Diagonalisation

Sample practice

Find eigenvalues of [[2,1],[1,2]]
Diagonalise a 2×2 matrix

08 OrthogonalityWork with perpendicular vectors and projections.

Key concepts

Orthogonal sets
Projections
Gram–Schmidt process

Sample practice

Project (3,4) onto (1,0)
Orthogonalise two vectors

09 Least SquaresFit models when equations have no exact solution.

Key concepts

Normal equations
Linear regression as least squares
QR factorisation

Sample practice

Fit a line to four points
Explain why least squares minimises error

10 Applications and SVDUse linear algebra in practice.

Key concepts

Markov chains
Singular value decomposition
Principal component analysis

Sample practice

Find the steady state of a Markov chain
Explain PCA in words

Use AI as your personal tutor, not an answer machine

The course book includes ready-to-use prompts for ChatGPT, Copilot, Gemini or any capable AI assistant. Each mode makes the AI teach, guide, check or challenge you, so you build real understanding.

1TEACH ME

Explain [topic] in Linear Algebra to me step by step at a [beginner / intermediate / advanced] level. Use a simple example first, then a harder one, and ask me a question to check I understood.

2GUIDE ME

I am working on this problem: [paste problem]. Do not give me the answer. Give me one hint at a time and wait for my next step.

3CHECK MY WORK

Here is my solution: [paste working]. Check each step, tell me exactly where any mistake is and why, but let me correct it myself.

4CHALLENGE ME

Give me three exam-style Linear Algebra questions on [topic], getting harder each time. Wait for my answers, then mark them and explain what I missed.

5HELP ME APPLY IT

Show me how [topic] from Linear Algebra is used in a real situation, then give me a small task to try it myself.

Verify before you trust

Ask a clear, specific question
Solve it yourself first
Test the answer by substituting or estimating
Check it against the worked examples in the book
Validate with your teacher if the AI and the book disagree

Don’t just study it. Build evidence you can show.

Project 1

Computer Graphics Transformations

Build rotation, scaling and shear matrices and apply them to a shape.

Project 2

Network and Markov Model

Model a simple network or market share with a Markov chain.

Project 3

Regression by Least Squares

Fit and evaluate a model with the normal equations.

Project 4

Image Compression with SVD

Explore how SVD approximates an image.

Final capstone

Linear Algebra Applications Portfolio

Combine worked problems and three applications with written explanations.

Who this course book is for

First and second-year university students
Engineering and physics students
Computer science and data science students
Economics students
Self-learners preparing for machine learning
Teachers refreshing the subject

Where these skills lead

Data science
Machine learning
Engineering
Computer graphics
Quantitative finance
Research

The course builds skills and evidence; it does not guarantee employment or exam results.

What you receive

10 progressive learning modules
Learning objectives, key concepts and practice problems
Ready-to-use AI Tutor prompts for five learning modes
Practical projects and a capstone
Portfolio guide, assessment rubric and completion checklist
Instant download and lifetime access in My Courses

Format	Digital course book (PDF)
Learning mode	Self-paced + AI Tutor
Level	Beginner → Advanced
Study time	About 40 hours
Created by	NoteStore Academy
Certificate	Optional project and portfolio assessment
Licence	Personal learning use
Updated	November 3, 2026

Certificate pathway

Want a certificate? Submit your work for optional assessment by NoteStore Academy. Certificates reflect demonstrated work, so they are not issued automatically with purchase.

Module practice work20%

Projects 1–440%

Capstone project30%

Portfolio and reflection10%

Frequently asked questions

Is calculus required?

No. Linear algebra stands on its own. Basic algebra is enough to start.

Does it include proofs?

Yes, at an introductory level, with guidance on how to read and write them.

Is it useful for machine learning?

Yes. Least squares, eigenvalues, SVD and PCA are central to machine learning and are covered in modules 7–10.

Do I need software?

No, but optional NumPy and online calculator tips are included.

How do I get the course book after paying?

Your download link appears on the order confirmation page straight away, is emailed to you, and stays in My Courses in your account.

Which AI assistant can I use?

Any capable assistant works, such as ChatGPT, Microsoft Copilot or Google Gemini. The prompts are written to work with all of them.

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