Chapter 2 — Further Reading
Annotated pointers for going deeper on vectors, addition, scalar multiplication, and magnitude. Mapped to the three standard texts this book tracks (Strang, Axler, Boyd–Vandenberghe), plus free resources. Section numbers are approximate and edition-dependent — use them as a guide, not gospel.
Core textbooks
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Gilbert Strang, Introduction to Linear Algebra (5th/6th ed.), §1.1–§1.2. Strang opens the entire book with vectors, linear combinations, and lengths, in the same geometry-first spirit as this chapter. His §1.1 ("Vectors and Linear Combinations") is the closest match to our §2.2 and §2.4; §1.2 introduces lengths and the dot product (our magnitude, plus a preview of Chapter 18). If you read one outside source for this chapter, read this — the worldview is the same. Strang's free MIT OpenCourseWare course (18.06) covers the same ground in video.
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Sheldon Axler, Linear Algebra Done Right (4th ed.), §1A–§1B. Axler is the abstract/algebraic counterpart: he treats $\mathbb{R}^n$ and $\mathbb{C}^n$ and moves quickly toward the vector-space axioms (our §2.4 preview and Chapter 5). He deliberately downplays the geometric picture and determinants, so it is the right place for math majors who want the rigorous, coordinate-free attitude early — but pair it with Strang for intuition. Note Axler's vectors are not committed to being columns; he works abstractly from the start.
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Stephen Boyd & Lieven Vandenberghe, Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares (free PDF). Chapters 1–3 are the applied treatment of exactly this chapter's content: vectors as data, vector addition and scalar multiplication, linear combinations, and norms/distance, all motivated by real applications (feature vectors, time series, document word counts). Best fit for CS / data science readers — it formalizes the "vectors as data" and word-count examples from our §2.9 and Case Study 2.2. Freely downloadable from the authors' site, with a companion Julia/Python language guide.
Video and visual
- 3Blue1Brown, Essence of Linear Algebra, Chapter 1 ("Vectors, what even are they?") and Chapter 2 ("Linear combinations, span, and basis vectors"). Grant Sanderson's animations are the single best visual companion to this chapter. Chapter 1 nails the arrow-vs-list duality (§2.1) and the physics/CS/math perspectives on what a vector "is"; Chapter 2 animates linear combinations and span exactly as our §2.4 and §2.6 describe, and sets up Chapter 6 of this book. Watch both before or after reading — they reinforce the geometry-first method directly. Free on YouTube.
Free online and interactive
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Khan Academy, "Vectors" (Linear Algebra unit). Gentle, exercise-rich coverage of vector addition, scalar multiplication, components, and magnitude, with immediate practice problems. Good for shoring up mechanics if the hand computations in our exercises feel shaky. Free.
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Immersive Math, Immersive Linear Algebra (Chapters 1–2). A free online book with interactive figures you can drag — particularly good for feeling the parallelogram rule (§2.2) and scalar multiplication (§2.3) in your hands rather than just reading them. Free.
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numpy documentation:
numpy.linalg.normand array broadcasting. For the computational track, skim the officialnumpy.linalg.normreference (note theordargument previewed in our Computational Note — the default is the Euclidean length) and the broadcasting rules (whyc * np.array(...)scales every component). Free.
A note on where to go next
- For the systems angle on linear combinations ("which weights hit the target?"), Chapter 3 of this book and Strang §1.3–§2.2 are the next step.
- For the abstract angle (why polynomials and functions are vectors too), Chapter 5 of this book and Axler §1B–§1C.
- For the geometry of length and angle that this chapter deliberately left informal, Chapter 18 of this book, Strang §1.2 and §4.1, and Boyd–Vandenberghe Chapter 3.