Chapter 3 Quiz — Systems of Linear Equations
Twelve conceptual checks. Try each before opening the answer. These test understanding of what a system means and what its solutions look like — not the solving algorithm (Chapter 4).
Q1. A system of linear equations can have how many solutions?
Answer
Exactly three possibilities: **none, exactly one, or infinitely many**. No linear system has exactly two (or any other finite number greater than one). This trichotomy is a consequence of linearity — geometrically, the solution set is always a flat object (a point, line, plane, …), never a scattered finite set of points.Q2. In the row picture of a two-equation, two-unknown system, what is each equation, and what is the solution set?
Answer
Each equation is a **line** in the plane; the solution set is the **intersection** of the lines. Crossing lines → one solution; parallel distinct lines → none; coincident lines → infinitely many. (In three unknowns, each equation is a *plane* and the solution set is the intersection of planes.)Q3. In the column picture, what question does solving $A\mathbf{x} = \mathbf{b}$ ask?
Answer
It asks: **is $\mathbf{b}$ a linear combination of the columns of $A$, and if so, with what weights?** The weights *are* the solution $\mathbf{x}$. A solution exists exactly when $\mathbf{b}$ lies in the span of the columns (the *column space* of Chapter 13).Q4. A system has no solution. What is it called, and what does the row picture say is happening?
Answer
It is **inconsistent**. In the row picture the surfaces share no common point (e.g. parallel distinct lines, or three planes forming a triangular prism). In the column picture, $\mathbf{b}$ lies *outside* the span of the columns — no combination of them can build it.Q5. Algebraically, what tell-tale equation appears when a system is inconsistent?
Answer
A contradiction with the unknowns gone — a row that reads $0 = c$ for some nonzero constant $c$ (e.g. $0 = 3$). Equivalently, $\operatorname{rank}([A\mid\mathbf{b}]) > \operatorname{rank}(A)$: appending $\mathbf{b}$ raised the rank, meaning $\mathbf{b}$ carries information the coefficient rows cannot.Q6. A consistent system in $n$ unknowns has $\operatorname{rank}(A) = r < n$. How many solutions, and how many free variables?
Answer
**Infinitely many** solutions, with $n - r$ **free variables**. The solution set has dimension $n - r$: one free variable → a line, two → a plane, and so on. (Zero free variables, $r = n$, would give a single point.)Q7. What does np.linalg.solve(A, b) return when $A$ is singular (inconsistent or dependent)?
Answer
It raises `LinAlgError: Singular matrix` in **both** cases. `np.linalg.solve` only handles the unique-solution case and cannot distinguish "no solution" from "infinitely many." To tell them apart, compare $\operatorname{rank}(A)$ with $\operatorname{rank}([A\mid\mathbf{b}])$, or use `np.linalg.lstsq`.Q8. For a square system $A\mathbf{x} = \mathbf{b}$, the determinant of $A$ is nonzero. What can you conclude about the solutions?
Answer
$\det(A) \neq 0$ means $A$ is invertible (full rank), so there is **exactly one solution for every** $\mathbf{b}$. If instead $\det(A) = 0$, the system has either no solution or infinitely many — never exactly one. (Determinant: Chapter 11; inverse: Chapter 9.)Q9. Why can a homogeneous system $A\mathbf{x} = \mathbf{0}$ never be inconsistent?
Answer
Because $\mathbf{x} = \mathbf{0}$ always satisfies it: $A\mathbf{0} = \mathbf{0}$. This is the **trivial solution**. The only open question for a homogeneous system is *uniqueness* — whether the trivial solution is the only one (columns independent) or whether nonzero solutions exist (columns dependent, a whole subspace of solutions — the *null space*).Q10. "More equations than unknowns means no solution." True or false?
Answer
**False.** Three lines all passing through a single common point form an over-determined system ($3$ equations, $2$ unknowns) with exactly one solution. What governs the solution count is the **rank** and the consistency of $\mathbf{b}$, not the head-count of equations versus unknowns.Q11. Two planes in $\mathbb{R}^3$ are given. Can the system of those two equations have exactly one solution?
Answer
**No.** Two planes are parallel-distinct (no solution), coincident (a plane of solutions), or cross in a *line* (infinitely many). They can never meet in a single point — that needs a third, suitably tilted plane. So two equations in three unknowns give either zero or infinitely many solutions.Q12. PageRank treats "rank the web" as what kind of mathematical object, and where is it actually solved in this book?