Chapter 5 Exercises — Vector Spaces

How to use these. Work the ⭐ problems first to lock in the definition and the eight axioms; they need no computation. The ⭐⭐ problems are short by-hand verifications — mostly "is this a vector space (or subspace)?" decisions where you must check closure and the zero vector. The ⭐⭐⭐ problems are proofs from the axioms (math-major core) and light numpy experiments. The ⭐⭐⭐⭐ problems connect vector spaces to real applications. Tags: [hand] = pencil only, [proof] = write a rigorous argument from the axioms, [code] = needs numpy, [essay] = written argument. For every "is it a vector space?" problem, state the operations you are assuming, then either verify the load-bearing axioms or exhibit a single counterexample.

Tier ⭐ — Conceptual (what is / why)

5.1 [hand] State all eight vector space axioms from memory, grouping them into the two families described in Section 5.3 ("addition is well-behaved" vs. "scaling cooperates with addition"). For each axiom, give the one-phrase plain-English meaning.

5.2 [hand] In your own words, explain the chapter's threshold idea that "vector is a role, not a kind of object." Name three objects that are vectors but are not arrows, and for each say what "addition" and "scalar multiplication" mean.

5.3 [hand] Which single axiom is the chapter's "load-bearing" one — the one that fails most often when a candidate set is not a vector space? Explain why the other axioms are usually inherited "for free" when you are looking at a subset of a known vector space.

5.4 [hand] Explain the difference between the two distributive axioms, $c(\mathbf{u}+\mathbf{v}) = c\mathbf{u}+c\mathbf{v}$ and $(c+d)\mathbf{v} = c\mathbf{v}+d\mathbf{v}$. Give a one-sentence reason the definition needs both rather than deriving one from the other.

5.5 [hand] The zero function $z(x) = 0$ and the zero polynomial $0 + 0x + 0x^2$ are both "zero vectors." Why does Theorem 5.1 let us call each of them the zero vector (with a definite article) rather than a zero vector?

5.6 [hand] Why is the field of scalars part of the data of a vector space? Give the chapter's example of one set that is a vector space over two different fields with two different dimensions.

5.7 [hand] Explain in two or three sentences why proving a theorem "for any vector space" is more valuable than proving it for $\mathbb{R}^n$. Use the uniqueness of the zero vector (Theorem 5.1) as your example.

5.8 [hand] True or false, with a one-line reason each: (a) every vector space contains a zero vector; (b) the additive inverse $-\mathbf{v}$ and the scalar multiple $(-1)\mathbf{v}$ are always the same vector; (c) the additive inverse of a matrix $A$ in $M_{2\times2}$ is its matrix inverse $A^{-1}$; (d) a set can be a vector space under one choice of operations and fail to be one under another.

Tier ⭐⭐ — Is it a vector space / subspace? (decide by hand)

For each, assume the standard operations unless told otherwise (componentwise for tuples, coefficientwise for polynomials, entrywise for matrices, pointwise for functions). Decide yes/no, and justify: if yes, note that closure and the zero vector hold; if no, give one explicit counterexample.

5.9 [hand] Is the set of all vectors $(x_1, x_2) \in \mathbb{R}^2$ with $x_2 = 3x_1$ a subspace of $\mathbb{R}^2$? (A line through the origin.)

5.10 [hand] Is the set of all $(x_1, x_2) \in \mathbb{R}^2$ with $x_1 x_2 = 0$ (the union of the two axes) a subspace? Hint: try adding $(1,0)$ and $(0,1)$.

5.11 [hand] Is the set of all $(x_1, x_2, x_3) \in \mathbb{R}^3$ with $x_1 + x_2 + x_3 = 0$ a subspace of $\mathbb{R}^3$? What about $x_1 + x_2 + x_3 = 1$?

5.12 [hand] Is the set of polynomials in $\mathbb{P}_2$ of degree exactly $2$ (so $a_2 \ne 0$) a vector space? Hint: is the zero polynomial in the set? Is $(x^2) + (-x^2)$ in the set?

5.13 [hand] Is the set of polynomials $p \in \mathbb{P}_3$ with $p(0) = 0$ a subspace of $\mathbb{P}_3$? Is the set with $p(0) = 2$?

5.14 [hand] Is the set of $2\times 2$ matrices with $\det(A) = 0$ a subspace of $M_{2\times 2}$? Hint: add $\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$ and $\begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}$ and check the determinant of the sum.

5.15 [hand] Is the set of symmetric $2\times 2$ matrices (those with $A^{\mathsf{T}} = A$) a subspace of $M_{2\times 2}$? Is the set of matrices with trace $0$?

5.16 [hand] Is the set of all continuous functions $f$ on $[0,1]$ with $f(0) = f(1)$ a subspace of $\mathcal{F}[0,1]$? Is the set with $f(0) = 1$?

5.17 [hand] Is the set of all real-valued functions $f$ on $\mathbb{R}$ with $f(x) \ge 0$ for all $x$ (the nonnegative functions) a vector space under pointwise operations? Probe a negative scalar.

5.18 [hand] Consider $\mathbb{R}^2$ with nonstandard operations: ordinary addition, but scalar multiplication redefined as $c \odot (x_1, x_2) = (c x_1, 0)$. Which axiom fails? Hint: check Axiom (5), $1\odot \mathbf{v} = \mathbf{v}$.

Tier ⭐⭐⭐ — Proofs from the axioms (A) and numpy experiments (C)

5.19 [proof] Prove that the additive inverse of a vector is unique: if $\mathbf{v} + \mathbf{w} = \mathbf{0}$ and $\mathbf{v} + \mathbf{w}' = \mathbf{0}$, then $\mathbf{w} = \mathbf{w}'$. Use only the axioms, and model your write-up on the four-part format of Theorem 5.1.

5.20 [proof] Prove that $-\mathbf{v} = (-1)\mathbf{v}$ in any vector space, citing each axiom you use. (Section 5.8 sketches the one-line argument; write it out fully, stating where Theorem 5.2 enters.)

5.21 [proof] Prove the "mirror" of Theorem 5.2: for any scalar $c$, $c\,\mathbf{0} = \mathbf{0}$. (The Check Your Understanding in Section 5.5 hints at the distributive law to use.)

5.22 [proof] Prove the cancellation law for vector addition: if $\mathbf{u} + \mathbf{w} = \mathbf{v} + \mathbf{w}$, then $\mathbf{u} = \mathbf{v}$. Which axiom guarantees the $-\mathbf{w}$ you add to both sides exists?

5.23 [proof] Prove that if $c\,\mathbf{v} = \mathbf{0}$ for a scalar $c$ and vector $\mathbf{v}$, then either $c = 0$ or $\mathbf{v} = \mathbf{0}$. Hint: if $c \ne 0$, multiply both sides by $1/c$ (the field lets you) and use Axioms (5), (6). State clearly where you use that the scalars form a field.

5.24 [proof] Prove that the intersection of two subspaces $U$ and $W$ of a vector space $V$ is itself a subspace. (Check closure under addition and scalar multiplication, and that $\mathbf{0}$ is in both.) Then give a concrete example in $\mathbb{R}^3$ showing the union of two subspaces is usually not a subspace.

5.25 [code] Implement the chapter's is_closed_under_combination(vectors, scalars, candidate_test) experiment in plain Python (see the Build Your Toolkit callout). Run it on (a) the half-plane $\{w : w_0 \ge 0\}$ with scalars including $-1$, and (b) the line $\{w : |w_1 - 2w_0| < 10^{-9}\}$ through the origin. Confirm it returns False for (a) and True for (b). In a comment, explain why a True result is evidence but not proof while a False result is a genuine disproof.

5.26 [code] Sample three functions $f(x) = 1$, $g(x) = \cos(2\pi x)$, $h(x) = \cos(4\pi x)$ on a 100-point grid over $[0,1]$ with np.linspace. Form the combination $\mathbf{w} = 2f - g + 0.5h$ two ways — "combine the formulas, then sample" and "sample each, then combine the vectors" — and verify with np.allclose that they agree. Which axioms is this experiment illustrating?

Tier ⭐⭐⭐⭐ — Applications and synthesis

5.27 [essay] Signals as vectors. A noise-cancelling headphone measures the ambient sound $a(t)$ and plays a counter-signal so that the listener hears $a(t) + c(t) \approx 0$. In 150–250 words, explain why this only makes sense because sounds live in a vector space: identify the addition, the role of the zero vector, and the additive inverse, and explain why the counter-signal $c(t)$ is essentially $-a(t)$. Which axiom guarantees that $-a(t)$ exists in the space?

5.28 [essay] Curve design. Re-read the Bézier-curve Real-World Application in Section 5.7. In 150–250 words, explain why a graphics program can guarantee that "the blend of two cubic curves is still a cubic curve," and connect this to the closure axiom. What is the "vector," what are the "scalars," and what plays the role of coordinates?

5.29 [essay] The qubit. In 150–250 words, explain in what precise sense a qubit's state $\alpha\,\mathbf{e}_0 + \beta\,\mathbf{e}_1$ is a vector in a vector space, why that space is complex (over $\mathbb{C}$) rather than real, and how the word "superposition" connects this chapter to the linearity of Chapter 1. Name the three later chapters (by number) that develop the quantum thread, and the name of the infinite-dimensional space that appears when a particle's position can be any real number.

5.30 [proof/essay] Why abstraction pays. Pick one structural fact proved in this chapter (uniqueness of zero, $0\mathbf{v} = \mathbf{0}$, or $-\mathbf{v} = (-1)\mathbf{v}$) and write it out as it instantiates in four different spaces — $\mathbb{R}^3$, $\mathbb{P}_2$, $M_{2\times 2}$, and $\mathcal{F}[0,1]$ — giving the concrete object in each case. Then argue in 100–150 words why the abstract proof is preferable to proving the fact four separate times.