Getting Started With Linear Algebra Done Right
Most people hit a wall when they try to learn linear algebra from the usual undergraduate textbooks. They get buried under matrix computations before they ever understand what a vector space actually is. Sheldon Axler's Linear Algebra Done Right takes the opposite route, which is why it exists and why people keep recommending it despite its quirks. The book skips determinants until the third chapter. Not because they don't matter, but because Axler thinks they distract from the core structure of linear algebra. You learn about operators, eigenvalues, and inner product spaces before you ever compute a determinant by cofactor expansion. It feels strange at first if you are used to the traditional sequence. By chapter three, that decision starts to make sense. Determinants become a consequence of the theory rather than the foundation you build everything on top of.
Working Through Linear Algebra Done Right
I worked through this book when I was trying to get serious about numerical methods for a machine learning project I was building around 2019. The standard approach at the time was to skip the proofs and just learn the matrix multiplication recipes, but I kept running into situations where understanding the operator structure mattered. Specifically, I was dealing with a generalized eigenvalue problem involving a nearly singular matrix, and the usual numerical routines were blowing up because I didn't have a clear picture of what the underlying operator was actually doing. Axler's treatment of the spectral theorem helped me see that I could decompose the problem differently. Instead of trying to invert a messy matrix, I restructured it using the orthogonal decomposition that the theorem guarantees. The workaround was straightforward once I understood the operator theory: project onto the eigenspaces, handle each one separately, and recombine. That took me from two days of debugging unstable code to about four hours of clean implementation. I should note that this only works well when your matrix is symmetric or normal. If it isn't, you are back to standard numerical linear algebra practices. The proofs are where most people either click or click away. They are direct but not hand-wavy. You will see arguments like the one showing that every operator on an odd-dimensional real vector space has an eigenvalue, which rests entirely on the fundamental theorem of algebra applied to the characteristic polynomial. It is elegant but it also reveals a limitation: this proof technique simply does not generalize to even dimensions over the reals. That is why Axler introduces complex vector spaces later, and why some readers find the early chapters frustratingly restricted.
One thing the book does not emphasize enough is computational practice. It will not teach you how to implement Gram-Schmidt efficiently on actual hardware. I ran into this gap when I needed to orthogonalize a set of vectors in a high-dimensional space for a recommendation system. The textbook derivation of the modified Gram-Schmidt process assumes infinite precision. In floating point arithmetic, you lose orthogonality faster than the book implies, and you end up with a basis that is numerically unstable. The fix is to use reorthogonalization — run Gram-Schmidt twice over the same vectors. It costs roughly double the operations but keeps your condition number manageable for most practical sizes under a few thousand dimensions. Another counter-intuitive point worth noting: Axler avoids determinants not just for pedagogical reasons but because they do not scale well to infinite-dimensional spaces. If you ever move into functional analysis or Hilbert space theory, the operator-theoretic perspective you build here translates directly. The determinant-based approach hits a hard ceiling. This is not a minor detail. It is the reason graduate-level courses in spectral theory start exactly where this book leaves off. There are legitimate downsides to this approach. The omission of determinants in the early chapters means you will struggle if your immediate goal is to pass a standard undergraduate exam that expects cofactor expansions and Cramer's rule. The book also provides very few computational exercises. The problem sets lean heavily toward proof construction, which is valuable but not helpful if you need to code these methods quickly. I found myself switching to Trefethen and Bau for the numerical side while keeping Axler for the theoretical framework.
Get the Full Details
Another bottleneck appears when you reach the chapter on singular value decomposition. Axler derives it from the spectral theorem for self-adjoint operators, which is correct but abstract. If you need to understand SVD from a computational perspective — say, for dimensionality reduction or low-rank approximation — you will want a supplement that covers the practical algorithms like Golub-Reinsch. The theoretical derivation takes about six pages. The actual implementation considerations span several chapters in other texts. The best way to use this book is to pair it with exercise work. Do the proofs yourself before reading the solutions. The ones on subspaces, quotient spaces, and the structure of operators are where the book earns its reputation. Skip the determinant-heavy sections if you are using it as a primary text, but do not skip them entirely if you need determinants for something like computing volumes or Jacobians in applied work. They return in chapter three and are handled with more maturity than in most first courses. I have seen people claim this book makes linear algebra easier. It does not make it easier. It makes it clearer, which is different. The clarity comes at the cost of computational preparedness. If you need both, use this as your second pass after a more computational introduction, or alongside one. A single semester course built around this text without supplementary numerical work will leave students who can prove things but cannot implement them.
The errata online for the third edition is reasonably maintained. There are a handful of typos in the earlier printings, mostly in notation consistency. If you are reading the third edition from a PDF, check the author's website for the corrections list before you get confused by an inconsistent symbol. It saves maybe ten minutes, but it saves them at points where you are already frustrated. If you are working through this alone, plan on spending about ten to fifteen hours per chapter for the first four chapters if you are doing the exercises properly. The later chapters move faster because the machinery is in place. The inner product spaces chapter, which is roughly forty pages, typically takes longer than the generalized operator chapters that follow because it requires rebuilding your intuition about geometry from scratch. That is normal. It is not a sign that you are doing it wrong.
