Immersive Linear Algebra (2016)
Upvoting this because I think it&#x27;s really important that we start getting more open source textbooks which specialize in exposition[1]. That being said, I&#x27;d make a different arrangement of the material.<p>Here are a few thoughts:<p>1. This book embraces the modern take on a rigorous course in linear algebra, which I disagree with. That is to say, it covers vector spaces front and center in the first chapter. Conversely, systems of linear equations (Gaussian elimination) and corresponding notions like row reduction of coefficient&#x2F;augmented matrices are pushed back several chapters. You see this in popular &quot;flagship&quot; textbooks like Axler&#x27;s <i>Linear Algebra Done Right</i> because they&#x27;re prioritizing the underlying theory. I think Gaussian elimination should be covered first because it motivates the material - you&#x27;re never going to cover Linear Algebra in a way that makes all the pieces fall together in a purely straightforward dependency.<p>But if you first cover systems of linear equations, you actually motivate the purpose of vector spaces. Moreover, many (most?) proof-based problems in linear algebra reduce to solving a system of linear equations (and therefore reducing a matrix) which is modeled after the thing you&#x27;re trying to prove. What&#x27;s the basis of a null space of this linear transformation? You can solve it with row reduction. Is this spanning set a basis of this vector space? Solvable through row reduction. You get a lot of equipment for solving vector space problems if you build up the row reduction scaffolding to a minimal level before vector space coverage. This does not have to temper the theoretical rigor of later, abstract material; if anything it grounds and augments it.<p>This is why I&#x27;m a huge fan of the way that Hoffman &amp; Kunze cover this material. That is a <i>heavily</i> theoretical book, which easily surpasses undergrad-level linear algebra in later chapters. A couple of decades ago it had the spotlight as the really good linear algebra book (it displaced Halmos&#x27; <i>Finite Dimensional Vector Spaces</i>, and was itself displaced by Friedberg or Axler). But it motivates a very rigorous coverage of vector spaces by starting with Gaussian elimination, matrix representation of systems of linear equations and (reduced) row echelon forms. If you don&#x27;t cover this first, you have to either assume the student knows it already or you resign yourself to wading through several chapters of potentially unmotivated abstraction.<p>2. In my opinion, linear maps should be covered directly after vector spaces. They&#x27;re a logical succession of the same concept - after you&#x27;ve taught linear combinations, linear (in)dependence, spanning sets, bases, subspaces, and related theorems, it&#x27;s very easy to jump right into functions <i>between</i> vector spaces. It&#x27;s just an expansion on all of these things. I would therefore rather see chapter 9 immediately following chapter 2. And much like my prior point, linear maps are an abstraction that empowers a lot of the other ideas; this is mainly because so many things are representable as a linear transformation (much like they&#x27;re representable as a matrix).<p>3. The rank is a very important concept, but I would argue it should be subsumed into the chapter on matrices. You can cover anything that doesn&#x27;t fit cleanly in the matrices chapter in the chapter on linear maps, since you&#x27;ll need to do that anyway before you cover the dimension theorem. That might be a good argument for simply covering matrix multiplication and addition in prerequisite material and subsuming all of matrices under Gaussian elimination and&#x2F;or linear maps too...<p>4. I&#x27;m a big fan of the prerequisite material covering basic trigonometry. Trigonometric functions are a good source of exercises for vector spaces. I would like to see an expansion which does minimal coverage of naive set theory (sets, subsets, unions&#x2F;intersections, etc), functions (injection, surjection, bijection) and polynomials. If you cover the basic definition and arithmetic of polynomials here, you save yourself some space when you cover Lagrange interpolation and factorization later on. It&#x27;s probably a lot to ask for, but some coverage of differentiation of polynomials would also be useful because the theory of vector spaces gets a lot richer when you incorporate exercises which use a bit of calculus&#x2F;analysis. In particular, using vector spaces of functions and linear maps of differentiation give you a lot of leeway to show how abstract vector spaces can be without going nuts on the moon math.<p>5. I&#x27;ve received pushback on this before, but I think it&#x27;s really important to cover fields as their own subject before covering vector spaces. If you&#x27;re going to cover vector spaces rigorously, you really need to cover the field axioms. Scalar multiplication doesn&#x27;t make sense without first defining a field (you can handwave it, but it&#x27;s easy to make subtle mistakes - like missing that the set of all integer $n$-tuples <i>cannot</i> comprise a vector space). Likewise you ordinarily need your linear map to agree on a field to be well-defined. This coverage would ideally also establish the basic facts of the real and complex fields, which are almost entirely the fields you work with in undergrad linear algebra.<p>6. I don&#x27;t see any exercises. A good math textbook challenges its reader to internalize each chapter&#x27;s material by substantially using it to prove further material. By not including exercises, this text becomes more of an illustrated monograph. That&#x27;s totally fine if it&#x27;s what they&#x27;re going for, but I think there&#x27;s a huge opportunity to tailor it even further with targeted exercises that expand on the theorems in each chapter.<p>Note that I&#x27;m mostly nitpicking here, even if I think my critiques have merit. The exposition of material is much more important than the arrangement of material, as long as things are building on each other rigorously. I&#x27;m not trying to say the textbook is bad; this is just a stream of consciousness about my opinions on arranging linear algebra for maximal pedagogy. You generally have to supplement textbooks in order to get a rich understanding of the subject matter anyway.<p>On the plus side, I&#x27;m a huge fan of the way they present vectors geometrically. That&#x27;s a huge win for illustration&#x2F;animation purposes. I do think there&#x27;s a missed opportunity to make a sexy illustration of transforming a system of linear equations into coefficient, unknown and constant matrices; then from there illustrating the transformation of the augmented matrix into its reduced row echelon form.<p>__________________________________<p>1. For pretty good open source, undergrad-level textbooks in Abstract and Linear Algebra respectively, see <a href="http:&#x2F;&#x2F;abstract.ups.edu&#x2F;aata&#x2F;" rel="nofollow">http:&#x2F;&#x2F;abstract.ups.edu&#x2F;aata&#x2F;</a> and <a href="http:&#x2F;&#x2F;linear.pugetsound.edu&#x2F;html&#x2F;fcla.html" rel="nofollow">http:&#x2F;&#x2F;linear.pugetsound.edu&#x2F;html&#x2F;fcla.html</a> respectively.