Recommendations on the number of exercises to do from Linear Algebra and Its Applications 6th edition (by Lay & McDonald)

Question

So I wanted to review my intro to Linear Algebra course in prep for next semester classes (starts Aug 2025/Fall since taking 2 gap semesters starting Dec 12th for personal reasons) like "a second course in Linear Algebra", and Linear Algebra and Its Applications 6th edition (by Lay & McDonald) is the textbook I'm deciding to use (since my uni uses it). I've already taken this class btw, but I feel I forgot many of the theorems, proofs, and computational methods we learned in this class (my weakest being the proofs for theorems and the "Symmetric Matrices and Quadratic Forms" topic).

Hence I feel I need to review it a bit more thoroughly by reviewing the Chps 1-7 + Chp 10 (which were the ones covered in our class). Since there's a bunch of exercises for each chapter (58 sections total and usually 50+ questions per section), what would y'all recommend for me in terms of which and how many textbook exercises I should do? Would there be an external question-bank/test-bank source y'all would recommend instead that would be more efficient but still comprehensive?

Obviously common sense would say to just try doing the problems that seem to address my weak-points, but it's kind of hard for me to figure that out. I kind of have the habit of wanting to do all the exercises but I fear that'll make it difficult/near impossible to finish reviewing all the chapters. I still want to review the chapters thoroughly/comprehensively tho. So if anyone could give me some advice on how I should go about it I'd be grateful. Ideally I want to review everything in span of 1-2 months (and I won't have much to do per day so I can spend at least 12 hours per day).

In fact, if anyone who worked through the Lay's Linear Algebra textbook could give me some advice that would be very helpful too (I have access to the Student Study Guide if y'all are wondering)!

Sorry if this is a dumb question I'm asking, but I just really need some advice here.

Answer 1

I teach regularly out of this book and love it. The exercises, in my opinion, have a great mix of basic computational, relatively straightforward conceptual, and more complicated conceptual. I don't have a specific recommendation, but my general recommendation is

• to work maybe 3-4 of the "basic computational" problems from each section (say problems 5, 11, 13, 15 for most sections, choosing odds, since the answers are provided so you can double check). Make sure you are comfortable with those.
• do as many of the True/False as you have time for -- though there are a few clunkers but most of the T/F are great -- they check vocabulary, your understanding of the assumptions of the various theorems, and their results. I'd also go further and whenever possible if you think the answer is False -- and the statement is amenable -- come up with a counter example or two (**)
• work 5-6 of the problems -- usually numbered high 20s to low/mid 30s -- that are more conceptual; you need practice with those general statements: If $A$ an an $n \times n$ matrix with linearly independent columns, does $A^2$ necessarily span $\mathbb{R}^n$? [Yes, since by the invertible matrix theorem, when $A$ is made of linearly independent columns, it is invertible, so $A^2$ is invertible since it is the product of two invertible matrices (namely $A$ and $A$) and again by the invertible matrix theorem, any invertible matrix of size $n \times n$ must span all of $\mathbb{R}^n.$]

Four extra points:

• (**) Counterexamples! Try as best as you can to build up good intuition on counterexamples to false statements: Does this hold for $T: V \to V$ defined by $T(\mathbf{v}) = \mathbf{0}$? What about the transformation $T:\mathbb{R}^3 \to \mathbb{R}^2$ defined by $T( (x_1,x_2,x_3)) = (x_1,0)$? My students often struggle keeping that "toolbox" of counterexamples handy, and often when looking for one, try something overly complicated first.
• Have conversations with yourself -- taking the example of the Invertible Matrix theorem from above: Can you explain to yourself why the transformation $\mathbf{x} \mapsto A\mathbf{x}$ is onto whenever $A$ is made of linearly independent columns?
  • Ok, but not just saying "this is statement (e) and that's statement (i) of the IVT.
  • Can you do it using an explanation about pivots? "To have linearly independent columns, there must be a pivot in each column, which, since $A$ i square means that there is a pivot in each column, which means that $A\mathbf{x} = \mathbf{b}$ is always consistent, which means that for each $\mathbf{b}$ there is an $\mathbf{x}$ which is mapped to it."
  • Can you do it without pivots, at least explicitly? "Since $A$ is made of $n$ linearly independent columns, the rank of $A$ is $n$; thus the column space is of dimension $n$, which means that the columns of $A$ form a basis for $\mathbb{R}^n$ and thus must span $\mathbb{R}^n$. Thus, our transformation is onto."
• Especially if the next class is expected to be more focused on abstract vector spaces: Try to clarify which theorems and ideas you understand only in their matrix form and which you can do for general linear transformations and vector spaces.
• Finally the theorems & proofs: Spend time thinking about their structure. If the claim is that some set is linearly dependent, then that means we are likely going to exhibit a set of scalars $c_1, c_2, ...$. Is that how this proof actually goes? (If not, then the proof might be somewhat "fancier" or use a "trick.") Can you see how one or more hypothesis in the theorem statement gave us the scalars we need? Etc. Again, only from my experience, but I have a lot of students that try to memorize proofs/conceptual problem solutions and so every problem to them is sort of brand new. The more you can break out of this, the better, I think. And, along the way, the best advice I ever got and still give to my students is to doubt every theorem statement. Make the proof really convince you -- try to come up with a counterexample and see why it fails.