Linear algebra: choose a route from equations to subspaces
A matrix calculation is more useful when you can say what its solution set means. Choose a first-course computational bridge or a proof-oriented continuation, then connect elimination, linear maps and geometric checks without reading three full books in parallel.
Study route and exercise: University Reading List editorial project. How we curate · Sources checked:
Scope and responsibility
An undergraduate route through systems, subspaces and linear maps. Hefferon means the fourth edition, Strang the sixth edition (2023), and Axler the fourth edition (2024 copyright, with separately revised online files). This is not a claim of full-book or classroom testing. The numerical examples are invented exact-data problems, not a fit to an empirical dataset; they establish no performance guarantee for numerical software.
Choose your starting point
I need a free first-course spine
Start with Hefferon for a gradual transition from elimination to proofs. Its complete answer book is useful after a genuine attempt; it should not become the first step of every problem.
I want matrices, geometry and applications together
Choose Strang when your course uses the sixth edition or its column-space perspective fits your goal. Use the selected samples to test that approach before assuming that videos replace the book or assessment.
I can compute and now need abstract reasoning
Use Axler as a proof-oriented second pass. Its free fourth edition removes a price barrier, but not the need to reason about definitions, examples and linear operators.
Compare the three books
Linear Algebra
Jim Hefferon · Edition: 4 · 2020
Free computational-to-proof bridge
- Choose it when
- You can manipulate algebra and want elimination and structural arguments developed together.
- Use another route when
- You need the exact assignments of a different prescribed text, or already want an operator-focused second course.
- Selected reading
- Fourth edition: Chapter One on systems; Two on vector spaces; Three on maps, including its optional orthogonal-projection route.
- Limits and trade-offs
- The author expects a semester of calculus in the audience, although the opening exercise here only needs algebra. Official worked answers cover the exercises; attempt and justify your own solution first.
Try before choosing
Reduce the guide’s matrix by elimination, write all solutions rather than one, and explain why your row operations preserve the solution set.
Check your work: A free parameter is named, not silently set to zero. Substituting the whole parameterized family into the original equations verifies completeness together with the rank argument.
Introduction to Linear Algebra
Gilbert Strang · Edition: 6 · 2023
Matrix and subspace perspective
- Choose it when
- You want to move between equations, columns and geometric approximation, with selected applications.
- Use another route when
- Your primary need is a free complete textbook or a proof-first development of abstract operators.
- Selected reading
- Sixth edition: Chapters 1–3 for vectors, equations and fundamental subspaces; Chapter 4 for orthogonality and least squares.
- Limits and trade-offs
- The official site supplies contents, samples and selected solutions, not a free full sixth-edition entitlement. Fifth-edition links on the same site do not establish sixth-edition numbering.
Try before choosing
Use the columns of A to explain which right-hand sides can be reached before computing an inverse or least-squares fit.
Check your work: Identify c=a+b and a nonzero vector orthogonal to every column. A residual outside the column space is evidence about consistency, not a licence to change the observations.
Linear Algebra Done Right
Sheldon Axler · Edition: 4 · 2024
Free proof-oriented continuation
- Choose it when
- You are ready to explain why kernel, image, basis and dimension statements are true.
- Use another route when
- You have not yet learned to solve small systems or need a purely algorithmic exercise bank.
- Selected reading
- Fourth edition: Chapters 1–3 for vector spaces and linear maps; Chapter 6 for inner products and orthogonal projection.
- Limits and trade-offs
- The author presents this as a second-course text. An open PDF is legally readable, but its BY-NC licence is not unrestricted commercial reuse permission; a revision date is not a new numbered edition.
Try before choosing
Interpret A as T:R³→R³ and prove the kernel–image dimension relation for this example using explicit bases.
Check your work: Give two independent image vectors and one kernel vector, prove they span the claimed spaces, and check rank 2 plus nullity 1 equals the domain dimension.
Check your preparation
Do I need calculus for the first example?
No. You need simultaneous equations, fractions, dot products and the meaning of a parameter. Hefferon’s complete course assumes a calculus background; the other selected routes have their own maturity expectations.
What does singular tell me?
A square singular matrix has no inverse. It does not say that every right-hand side is inconsistent: the right-hand side must be checked against the column space.
Can software settle the proof?
A symbolic or numerical result can help check arithmetic. You still need to state the field, dimensions, exact-versus-rounded inputs and why the proposed family includes every solution.
Six steps with a concrete output
Outputs marked ready: 0 / 6
These are your own study marks, not an assessment. They are not sent to this site.
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Keep two representations
Selected reading: Hefferon One.I or Strang Chapters 1–2.
Make this: Write the three equations and the same problem as a linear combination of columns. Label all dimensions.
Check your work: A is 3×3, x and b have three components, and the column coefficients are the unknowns, not the coordinates of b.
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Find all solutions
Selected reading: Hefferon One.I.2–3 or the selected systems route.
Make this: Eliminate, identify pivots and express every solution with one free parameter.
Check your work: Distinguish a particular solution from a kernel vector. Substitute both pieces separately.
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Explain the image
Selected reading: Hefferon Two–Three; Strang Chapter 3; Axler Chapter 3.
Make this: Produce bases for the image and kernel, and justify independence and spanning.
Check your work: A list of vectors is not a basis until both requirements are checked. Count dimensions using the domain, not the number of equations alone.
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Test consistency geometrically
Selected reading: Strang Chapter 3 or the maps route.
Make this: Find a nonzero left-null vector and use its dot product as a consistency check for two right-hand sides.
Check your work: A nonzero dot product refutes exact consistency. A zero dot product suffices here only after showing that this vector describes the entire one-dimensional orthogonal complement.
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Change to approximation
Selected reading: Strang Chapter 4; Hefferon’s projection section; Axler Chapter 6.
Make this: Solve the follow-up and give a projected target, residual and full least-squares solution family.
Check your work: Check Aᵀr=0 and report the residual norm. Do not call the projection the original observation.
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Write a reproducible model note
Selected reading: Revisit the chosen book’s systems/subspaces sections.
Make this: Write a one-page explanation of existence, uniqueness and approximation, including the role of exact input values.
Check your work: Another reader can reproduce your family and projection. Keep measurement-error claims separate from these stipulated arithmetic inputs.
Worked exercise
Original example: one dependent column, many exact solutions
Use the exact real matrix A with columns a=(1,1,0), b=(0,1,1), c=(1,2,1), and target d=(4,7,3). Find every x with Ax=d, the rank and nullity, and a structural consistency check.
Assumptions
All entries are invented exact real numbers. The letter b denotes the second column, while d is the target. There is no noise model or external data. Use Euclidean dot products only when discussing orthogonality.
- Since c=a+b, Ax=(x₁+x₃)a+(x₂+x₃)b. The independent columns a,b give rank 2. The target is 4a+3b, so consistency holds.
- Set x₃=t. Then x₁=4−t and x₂=3−t, giving x=(4,3,0)+t(−1,−1,1), t∈R. The kernel is spanned by (−1,−1,1), so nullity is 1.
- Let n=(1,−1,1). Its dot product with a,b,c is zero, and n·d=4−7+3=0. Because the column space has dimension 2 in R³, its orthogonal complement is exactly span(n).
Answer and interpretation
There are infinitely many exact solutions, not none: x=(4−t,3−t,t). Rank 2 and nullity 1 describe different properties; their sum is 3, the domain dimension. A determinant of zero cannot by itself decide consistency for a particular target.
Test the result
Substitute into each row: x₁+x₃=4, x₁+x₂+2x₃=7, x₂+x₃=3. Check that adding a kernel vector changes coefficients but not the output.
Change an assumption
Change one target coordinate: exact fit disappears
Keep A and the exact Euclidean setting, but replace the target by e=(4,8,3). Find the least-squares projected target, residual and all minimizing coefficient vectors. No measurement uncertainty is added.
- n·e=4−8+3=−1, so the new target is outside the column space and Ax=e has no solution.
- The orthogonal projection is p=e−[(n·e)/(n·n)]n=e+(1/3)n=(13/3,23/3,10/3). The residual r=e−p=(−1/3,1/3,−1/3) has squared norm 1/3.
- Because p=(13/3)a+(10/3)b, every minimizer is x=(13/3−t,10/3−t,t). The projected output is unique even though its coefficients are not.
Answer and interpretation
The best attainable output is p=(13/3,23/3,10/3), with residual norm 1/√3. There is a whole coefficient family producing p. Neither choosing t=0 nor reporting a small residual establishes unique coefficients or a correct empirical model.
Test the result
Verify Aᵀr=0 and p+r=e. The orthogonal residual gives the Pythagorean minimum over the column space; merely trying several coefficient vectors would not establish the global least-squares result.
Common mistakes
One answer is not all answers
Setting a free variable to zero produces a particular solution, not a proof of uniqueness.
An inverse is not the only route
A singular square matrix can still map many coefficient vectors to a consistent target.
A fit is not a repaired observation
Keep e, its projection p and residual r named separately. Noise assumptions require separate evidence.
Before moving on
- Produce an exact solution family and verify it in the original equations.
- Give bases, dimensions and the relevant orthogonality checks.
- Explain why a unique projection can have nonunique coefficient vectors.
Sources and edition checks
Sources checked:
- Hefferon: official fourth-edition text and contents
The official fourth-edition PDF establishes the systems/vector-spaces/maps route. Selected front matter and contents were checked, not a new full-book reading.
- Hefferon: independent study and worked answers
The author describes complete worked answers and recommends attempting exercises before consulting them.
- Strang: sixth-edition identity, contents and samples
The official sixth-edition page distinguishes current samples and contents from older fifth-edition links.
- Axler: fourth edition, access and intended level
The author identifies the fourth edition as a second-course text and provides authorized free versions; online revision dates are separate from the 2024 copyright.
These are proposed study activities, not official course assignments or a claim that every book has been read cover to cover.