Chapter 35 — Exercises

Work the conceptual tier first; it builds the vocabulary the rest depend on. Problems marked (proof) ask for a rigorous argument from the axioms; (code) problems should be checked against numpy, but the understanding must come first. Throughout, $\mathbb{P}_n$ is the space of polynomials of degree at most $n$, $D=\frac{d}{dx}$ is differentiation, and $[T]_{C\leftarrow B}$ is the matrix of $T$ from basis $B$ to basis $C$.

Tier 1 — Conceptual (⭐)

1.1 State the two axioms of a linear transformation. For each, name the corresponding rule of calculus that makes differentiation linear.

1.2 True or false, with a one-sentence reason each: (a) Every linear map sends $\mathbf{0}$ to $\mathbf{0}$. (b) The map $T(x)=2x+1$ on $\mathbb{R}$ is linear. (c) The matrix of a linear operator is the same in every basis. (d) The kernel of $T:V\to W$ is a subspace of $W$.

1.3 In one sentence each, say what the kernel and the image of a linear map are, and which classical Chapter 13 subspace each one generalizes.

1.4 Explain in plain English why the matrix of a linear map depends on the chosen bases. Use a non-mathematical analogy (e.g., units of measurement, or photographing an object from different angles).

1.5 What does it mean for two matrices to be similar? Name three quantities that similar matrices must share, and say why (one sentence).

1.6 Define an isomorphism. State the theorem that classifies finite-dimensional vector spaces, and complete the sentence: "Every $n$-dimensional real vector space is isomorphic to ______."

1.7 Differentiation $D:\mathbb{P}_3\to\mathbb{P}_3$ is neither injective nor surjective. Give the kernel and the image, and connect "not injective" to the "$+C$" of indefinite integration.

1.8 Why is the differentiation operator on $\mathbb{P}_n$ called nilpotent? State the smallest power of $D$ that is zero on $\mathbb{P}_3$, and explain it in one sentence of calculus.

Tier 2 — Computation by hand (⭐⭐)

2.1 Build the matrix $[D]_B$ of differentiation on $\mathbb{P}_2$ in the monomial basis $\{1,x,x^2\}$. Then apply it to the coordinate vector of $p(x)=4-x+3x^2$ and confirm the result is the coordinate vector of $p'(x)$.

2.2 Let $T:\mathbb{P}_2\to\mathbb{P}_2$ be $T(p)(x)=p(x)+p'(x)$. Find $[T]_B$ in the monomial basis by computing $T$ on each of $1,x,x^2$. Is $T$ invertible? (Look at the diagonal.)

2.3 Let $T:\mathbb{P}_3\to\mathbb{R}$ be $T(p)=p(2)$ (evaluation at $2$). Write its $1\times 4$ matrix in the monomial basis, find its rank and nullity, and describe the kernel as a set of polynomials.

2.4 For the differentiation matrix $[D]_B$ on $\mathbb{P}_3$, compute $[D]_B^2$ and $[D]_B^3$ by hand. Interpret each as a derivative operator (second derivative, third derivative), and confirm $[D]_B^4=0$.

2.5 Let $S:\mathbb{P}_1\to\mathbb{P}_2$ be $S(p)(x)=\int_0^x p(t)\,dt$. Build its $3\times 2$ matrix in the monomial bases. Verify by hand that $[D]\,[S]=I_2$ (where $[D]$ is differentiation on $\mathbb{P}_2$), illustrating the Fundamental Theorem of Calculus.

2.6 The matrix of an operator $T$ on $\mathbb{R}^2$ in the standard basis is $A=\left[\begin{smallmatrix}2&1\\0&3\end{smallmatrix}\right]$. Using the change-of-basis matrix $P=\left[\begin{smallmatrix}1&1\\0&1\end{smallmatrix}\right]$, compute the matrix $P^{-1}AP$ of the same operator in the new basis $\{(1,0),(1,1)\}$. Confirm the trace and determinant are unchanged.

2.7 Compute the matrix of $D$ on $\mathbb{P}_3$ in the factorial basis $\{1,x,\tfrac{x^2}{2},\tfrac{x^3}{6}\}$ by differentiating each basis vector. Confirm it is the clean shift (all $1$s on the superdiagonal) and is similar to the monomial-basis matrix.

2.8 Determine the rank and nullity of each map, and verify rank–nullity: (a) $D:\mathbb{P}_4\to\mathbb{P}_4$; (b) the evaluation map $p\mapsto(p(0),p(1),p(2))$ on $\mathbb{P}_3$; (c) the transpose operator on $M_{2\times 2}$.

Tier 3 — Proof (⭐⭐⭐, A track) and Coding (⭐⭐⭐, C track)

3.1 (proof) Prove directly from the axioms that for any linear map $T:V\to W$, the kernel $\ker T$ is a subspace of $V$. (Show it contains $\mathbf{0}$ and is closed under addition and scalar multiplication.)

3.2 (proof) Prove that a linear map $T$ is injective if and only if $\ker T=\{\mathbf{0}\}$. Use linearity to reduce $T(\mathbf{u})=T(\mathbf{v})$ to a statement about $\mathbf{u}-\mathbf{v}$.

3.3 (proof) Prove that the image $\operatorname{im}T$ is a subspace of the codomain $W$, again from the two axioms only.

3.4 (proof) Prove that if $T:V\to W$ is an isomorphism, then its inverse function $T^{-1}:W\to V$ is also linear. (Apply $T$ to both sides of the candidate equations and use injectivity.)

3.5 (proof) Prove the abstract Rank–Nullity theorem $\dim\ker T+\dim\operatorname{im}T=\dim V$ for finite-dimensional $V$, by extending a basis of $\ker T$ to a basis of $V$ and showing the images of the extension vectors form a basis of $\operatorname{im}T$. (Reconstruct the §35.8 argument in your own words; state where each axiom is used.)

3.6 (proof) Let $T:V\to V$ be a linear operator on a finite-dimensional space. Prove that $T$ is injective if and only if $T$ is surjective. (Use rank–nullity; note this fails in infinite dimensions — see 3.10.)

3.7 (proof) Prove that similar matrices have the same trace, using $\operatorname{tr}(AB)=\operatorname{tr}(BA)$. Then prove they have the same determinant.

3.8 (code) Write diff_matrix(n) (pure Python, nested lists) returning the $(n+1)\times(n+1)$ matrix of $D$ on $\mathbb{P}_n$ in the monomial basis. Verify against numpy that (a) it matches the hand matrix for $n=3$; (b) applied to several random integer coefficient vectors it reproduces numpy.polynomial.polynomial.polyder; (c) $D^{n+1}=0$ via numpy.linalg.matrix_power.

3.9 (code) Write matrix_of_map(T, basis_in, basis_out, coords_out) that builds the matrix of an abstract linear map: it applies the callable T to each input-basis vector and uses coords_out to express the result in the output basis, stacking the results as columns. Test it by recovering diff_matrix(3) and the evaluation-map matrix $\left[\begin{smallmatrix}1&0&0&0\\1&1&1&1\end{smallmatrix}\right]$.

3.10 (proof) (Infinite-dimensional caveat.) On the space of all polynomials $\mathbb{P}$ (no degree bound), show that differentiation $D$ is surjective but not injective, while integration $S$ is injective but not surjective. Explain why this does not contradict 3.6. (Which hypothesis of 3.6 fails?)

Tier 4 — Application (⭐⭐⭐⭐)

4.1 (code) (Linear ODE as an operator equation.) Consider the operator $L=D^2-3D+2I$ acting on $\mathbb{P}_4$, where $I$ is the identity. Build its matrix in the monomial basis from diff_matrix(4), and compute its kernel numerically. Compare the dimension of the kernel to the number of independent solutions of the differential equation $y''-3y'+2y=0$ that happen to be polynomials. (Hint: the true solution space is spanned by $e^x,e^{2x}$ — how many of those are polynomials? Reconcile with rank–nullity.)

4.2 (code) (Polynomial interpolation.) The evaluation map $\operatorname{ev}:\mathbb{P}_2\to\mathbb{R}^3$, $p\mapsto(p(x_0),p(x_1),p(x_2))$ at three distinct nodes is an isomorphism. Build its $3\times 3$ matrix (the Vandermonde matrix), verify it is invertible, and use its inverse to find the unique quadratic through $(0,1),(1,3),(2,7)$. Confirm the polynomial passes through all three points.

4.3 (code) (Encoding map, CS.) A linear $[6,3]$ code over the reals encodes a 3-vector message $\mathbf{m}$ as $G\mathbf{m}$ for a $6\times 3$ generator matrix $G$ of rank $3$. Build such a $G$, confirm the encoding map is injective (so distinct messages get distinct codewords) via its kernel, and identify the image as a 3-dimensional subspace of $\mathbb{R}^6$. Explain in terms of rank–nullity why an injective encoder must map into a higher-dimensional space.

4.4 (Conceptual synthesis.) The momentum operator in quantum mechanics is essentially $\frac{d}{dx}$ acting on wavefunctions. Using only this chapter's ideas, explain: (a) why momentum is a linear operator; (b) what "choosing a basis" corresponds to physically; (c) why the eigenvalues of such an operator — not its matrix entries — are the physically measurable quantities. (Tie to §35.7 on basis-independence of eigenvalues. See operators in quantum mechanics.)

4.5 (proof + code) (Trace as a linear functional.) Show that the trace $\operatorname{tr}:M_{n\times n}\to\mathbb{R}$ is itself a linear map. Find its kernel (the traceless matrices) and its image, and verify rank–nullity: $\dim\ker+\dim\operatorname{im}=n^2$. Confirm numerically for $n=3$ that the traceless matrices form an $8$-dimensional space.