Chapter 15 Exercises — Dimension, Basis, and Coordinates

Work the tiers in order; each builds on the last. ⭐ conceptual · ⭐⭐ hand computation · ⭐⭐⭐ proof (A) or coding (C) · ⭐⭐⭐⭐ application. A problem marked [proof] wants a rigorous argument; [code] wants a short numpy snippet (and, where noted, your toolkit/coordinates.py). Verify every coordinate computation by reconstruction.

⭐ Tier 1 — Conceptual

15.1. State the two conditions a set must satisfy to be a basis. For each condition, describe in one sentence the kind of failure that occurs if only that condition is dropped.

15.2. True or false, with a one-line reason each: (a) Any spanning set of $\mathbb{R}^3$ is a basis. (b) Any linearly independent set in $\mathbb{R}^3$ is a basis. (c) Every basis of $\mathbb{R}^5$ has exactly five vectors. (d) The zero vector can belong to a basis.

15.3. A plane through the origin in $\mathbb{R}^3$ has dimension . A line through the origin in $\mathbb{R}^{100}$ has dimension . The space $\{\mathbf{0}\}$ has dimension . The space $P_4$ of polynomials of degree $\leq 4$ has dimension .

15.4. Explain, in your own words, why the same vector can have different coordinate vectors. What stays the same when the basis changes, and what changes?

15.5. Why are the standard coordinates of $(7, -3, 5)$ equal to its entries $(7, -3, 5)$? Reference the standard basis in your answer.

15.6. A set of six vectors in $\mathbb{R}^4$ is handed to you. Without any computation, what can you immediately conclude about whether it is linearly independent? Whether it spans? Whether it is a basis?

15.7. Explain the difference between the number of entries a vector is written with and the dimension of the space it lives in. Use the anchor plane (a 2D plane in $\mathbb{R}^3$) as your example.

⭐⭐ Tier 2 — Hand computation

15.8. Determine whether each set is a basis for $\mathbb{R}^2$. If not, say which condition(s) fail. (a) $\{(1, 2), (2, 4)\}$ (b) $\{(1, 0), (0, 1), (1, 1)\}$ (c) $\{(3, 1), (1, 2)\}$ (d) $\{(0, 0), (1, 1)\}$

15.9. Find the coordinate vector of $\mathbf{v} = (7, 2)$ relative to the basis $\mathcal{B} = \{(1, 1), (1, -1)\}$ by solving $B\mathbf{c} = \mathbf{v}$ by hand. Verify by reconstruction.

15.10. Find $[\mathbf{v}]_{\mathcal{B}}$ for $\mathbf{v} = (1, 5, 3)$ and the basis $\mathcal{B} = \{(1,0,0), (1,1,0), (1,1,1)\}$. (Hint: the matrix is triangular — use back-substitution.) Verify by reconstruction.

15.11. The vector $\mathbf{w}$ has coordinate vector $[\mathbf{w}]_{\mathcal{B}} = (2, -3)$ relative to $\mathcal{B} = \{(2, 1), (1, 3)\}$. Find the standard coordinates of $\mathbf{w}$. (This is the easy direction — just compute $B\mathbf{c}$.)

15.12. Consider the plane $P = \operatorname{span}\{(1, 1, 0), (0, 1, 1)\}$ in $\mathbb{R}^3$. (a) Confirm these two vectors form a basis for $P$. (b) Find the coordinates of $\mathbf{v} = (2, 3, 1)$ relative to this basis, if $\mathbf{v} \in P$, by solving the $3\times 2$ system. If $\mathbf{v} \notin P$, show the system is inconsistent.

15.13. Find the dimension of $\operatorname{span}\{(1, 2, 3), (2, 4, 6), (1, 0, 1)\}$ in $\mathbb{R}^3$ by row reduction, and give a basis for the span.

15.14. Write the coordinate vector of the polynomial $p(t) = 5 - 2t + 4t^2$ relative to the monomial basis $\{1, t, t^2\}$ of $P_2$. Then find its coordinate vector relative to the basis $\{1, t, t^2 - 1\}$ of the same space. (Hint: write $t^2 = (t^2 - 1) + 1$.)

15.15. Give the dimension of each space and one basis for it: (a) the $2 \times 2$ symmetric matrices; (b) the $2\times 2$ matrices with trace $0$; (c) the plane $x + y + z = 0$ in $\mathbb{R}^3$.

15.16. A vector $\mathbf{v} = (3, 3)$ is given. Find its coordinates relative to (a) the standard basis, (b) $\{(1,1),(1,-1)\}$, (c) $\{(3,0),(0,3)\}$. Comment on what made (a) and (c) easy and (b) require a (small) computation.

⭐⭐⭐ Tier 3 — Proof (A) / Coding (C)

15.17. [proof] Prove that any basis of $V$ gives every vector a unique coordinate vector. (Reconstruct the existence-and-uniqueness argument from §15.4; state clearly where spanning is used and where independence is used.)

15.18. [proof] Prove that in an $n$-dimensional space, any linearly independent set of exactly $n$ vectors is automatically a basis (i.e. it must also span). You may cite the replacement theorem.

15.19. [proof] Let $\mathcal{B}$ be a basis of $V$ and $\mathbf{w} \in V$. Prove that $\{\mathcal{B} \cup \{\mathbf{w}\}\}$ is linearly dependent. (This shows a basis is a maximal independent set.)

15.20. [proof] Show that the coordinate map $\mathbf{v} \mapsto [\mathbf{v}]_{\mathcal{B}}$ is linear: prove $[\mathbf{u} + \mathbf{v}]_{\mathcal{B}} = [\mathbf{u}]_{\mathcal{B}} + [\mathbf{v}]_{\mathcal{B}}$ and $[c\,\mathbf{v}]_{\mathcal{B}} = c\,[\mathbf{v}]_{\mathcal{B}}$ directly from uniqueness of coordinates.

15.21. [code] Implement coordinates(v, basis) in toolkit/coordinates.py per the Build Your Toolkit callout: build $B$ with the basis vectors as columns, solve $B\mathbf{c} = \mathbf{v}$ using your gaussian_elimination from toolkit/linear_systems.py (no numpy in the implementation), and verify by reconstruction, raising a clear error if $\mathbf{v}$ is not in the span. Test it against np.linalg.solve(np.column_stack(basis), v) for the bases in 15.9 and 15.10.

15.22. [code] Write a numpy function is_basis(vectors) that returns True iff a list of vectors is a basis for $\mathbb{R}^n$ (where $n$ is the length of each vector). It should check the count equals $n$ and the rank equals $n$. Test it on the four sets in 15.8 (after padding to $\mathbb{R}^2$) and on a known $\mathbb{R}^3$ basis.

15.23. [code] For the plane $P = \operatorname{span}\{(2,1,0), (1,1,1)\}$ (the chapter anchor), use numpy to (a) build an orthonormal basis $\{\mathbf{q}_1, \mathbf{q}_2\}$ of $P$ via Gram–Schmidt, and (b) compute the coordinates of $\mathbf{v} = (4, 1, -2)$ in both the original basis and the orthonormal basis, confirming both reconstruct to $\mathbf{v}$. Reproduce the numbers from §15.8.

⭐⭐⭐⭐ Tier 4 — Application

15.24. [application] (Color spaces.) A pixel's color is the vector $(180, 120, 40)$ in the standard RGB basis $\{\mathbf{R}, \mathbf{G}, \mathbf{B}\}$. A device uses the alternative primary basis $\mathcal{P}$ whose vectors, in RGB coordinates, are $\mathbf{p}_1 = (1,1,0)$, $\mathbf{p}_2 = (1,-1,0)$, $\mathbf{p}_3 = (0,0,1)$. (a) Confirm $\mathcal{P}$ is a basis of $\mathbb{R}^3$. (b) Find the color's coordinate vector in $\mathcal{P}$ by solving the system (numpy allowed), and verify by reconstruction. (c) In one sentence, explain why this is a change of coordinates and not a change of color.

15.25. [application] (Degrees of freedom.) A planar robot arm has two rigid links joined by two rotating joints, anchored at the origin; its configuration is the pair of joint angles $(\theta_1, \theta_2)$. (a) What is the dimension of its configuration space? (b) The end-effector position lives in $\mathbb{R}^2$. If you add a third link and joint, the configuration space dimension becomes ___, but the end-effector still lives in $\mathbb{R}^2$ — explain what the extra degree of freedom buys you, and connect it to the idea of a non-trivial null space of motions (Chapter 14).

15.26. [application] (Molecular vibrations.) A nonlinear molecule of $N$ atoms has $3N$ position coordinates. Subtracting the $3$ translational and $3$ rotational degrees of freedom leaves the vibrational modes. (a) How many vibrational modes does a water molecule ($N = 3$, nonlinear) have? (b) A linear molecule has only $2$ rotational degrees of freedom — how many vibrational modes does carbon dioxide ($N = 3$, linear) have? (c) Phrase both answers as "the dimension of the space of internal vibrations is ___."

15.27. [application] (Feature redundancy in data.) A dataset has four feature columns, but column 3 equals column 1 plus column 2 (a derived feature someone added). (a) What is the dimension of the span of the four feature vectors? (b) Explain why fitting a linear model on all four features gives non-unique coefficients, tying it to the uniqueness theorem of §15.4. (c) Which column would you drop to restore a basis, and why does the choice of which to drop not change the span?