Linear Algebra I: Vectors, Matrices & Systems
A first university-level linear algebra course focused on vectors, matrices, row-reduction and linear systems, with geometric intuition and exam-style practice.
What you’ll learn
- Think of vectors in ℝ²/ℝ³ as arrows and data points
- Use matrices and row operations to solve linear systems
- Understand span, linear independence, basis and dimension
- Use determinants to detect invertibility and understand area/volume scaling
Curriculum
M1
Vectors, matrices & geometry — 3 lessons
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Vectors in ℝ²/ℝ³ & basic operations14m
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Matrices as arrays & matrix operations16m
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Linear combinations, span & geometric intuition16m
M2
Row reduction & linear systems — 3 lessons
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Row reduction & echelon form16m
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Solving linear systems & interpreting solutions16m
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Rank, consistency & pivot structure14m
M3
Vector spaces, subspaces & dimension — 3 lessons
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Vector spaces & subspaces (col/row/null)16m
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Linear independence, basis & dimension16m
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Rank–nullity theorem & consequences16m
M4
Determinants & geometric meaning — 3 lessons
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Determinants: definitions & properties16m
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Determinants as area/volume scaling14m
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Invertibility criteria via determinant & rank14m
Projects
- Solve small systems and visualise solutions as lines/planes in ℝ²/ℝ³
- Build a tiny Python toolkit for row-reduction and determinant computation
- Create a formula/intuition sheet for rank, nullity and solution sets
Prerequisites
- Comfort with high-school algebra and basic functions
- At least one semester of single-variable calculus (or concurrent)
- Optional but helpful: basic Python or another programming language
Who is this for?
- Students in a first Linear Algebra course (vectors & systems)
- Self-taught learners preparing for machine learning or further math
- Engineers and CS students who want geometric intuition for matrices
Outcomes
- Confident with vectors, matrices, Gaussian elimination and determinants
- Able to connect algebraic manipulations to geometric pictures in ℝ²/ℝ³
- Ready for eigenvalues, inner products and transformations in Linear Algebra II
Resources
- Starter project / template (ZIP)
- Setup & study checklists (PDF)
- Core formulas / syntax cheatsheet (PDF)
Tip: Right-click → “Save link as…” if your browser opens the file.
FAQ
Is this beginner-friendly?
Yes. We start from mental models and build up with guided practice and plenty of worked examples.
Will I need extra software?
We stick to free or standard tools. Any optional extras are clearly marked and have alternatives.
Do I get updates?
Yes—lifetime access with updates as the field, tools, and exam expectations evolve.