Chapter 1 Quiz — What Is Linear Algebra?
Twelve quick checks on the big ideas. Try each before opening the answer. These are conceptual — no calculator needed.
Q1. In one phrase, what does linear algebra study?
Answer
**Linear transformations of space (and the vectors and spaces they act on).** Determinants, eigenvalues, and the rest are machinery for understanding those transformations — they are not the point.Q2. A transformation $T$ is linear. Which two equations must hold for all vectors $\mathbf{u}, \mathbf{v}$ and all scalars $c$?
Answer
$T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$ (preserves addition) and $T(c\mathbf{v}) = c\,T(\mathbf{v})$ (preserves scalar multiplication). Together they are **superposition**: $T(c\mathbf{u} + d\mathbf{v}) = c\,T(\mathbf{u}) + d\,T(\mathbf{v})$.Q3. True or false: the function $y = 3x + 7$ is a linear transformation.
Answer
**False.** The constant $+7$ moves the origin (at $x=0$, $y=7\neq 0$), so it violates $T(\mathbf{0}) = \mathbf{0}$. It is **affine**, not linear. A linear map must pass through the origin.Q4. Geometrically, what do the two columns of a $2\times 2$ matrix represent?
Answer
The **images of the basis vectors**: the first column is where $\mathbf{e}_1$ (east) lands, the second column is where $\mathbf{e}_2$ (north) lands. Those two arrows determine the entire transformation.Q5. The visualizer shows a matrix turning the unit square into a parallelogram with a horizontal bottom edge and a slanted left edge, same area as the square. What transformation is this, and what is its determinant?
Answer
A **shear**. Its determinant is **1** — a parallelogram with the same base and height as the square has the same area, so the area-scaling factor is 1.Q6. A matrix has determinant 0. What does that say about the transformation, geometrically?
Answer
It **collapses area to zero** — it squashes the plane onto a line (or a point), losing a dimension. Such a transformation is **singular** (non-invertible): once distinct points are mapped to the same place, no rule can separate them again. Projection is the classic example.Q7. Which of these is the correct mental model of matrix–vector multiplication $A\mathbf{v}$? (a) A memorized row-times-column rule; (b) a weighted sum of the columns of $A$, with weights given by the components of $\mathbf{v}$; (c) both describe the same operation.
Answer
**(c).** Both are true and give the same numbers, but **(b)** is the meaning — superposition rebuilding the transformed vector from the transformed basis. The row-times-column rule **(a)** is correct arithmetic but, taken alone, hides the idea.Q8. A linear map sends $\mathbf{e}_1 \mapsto (2, 0)$ and $\mathbf{e}_2 \mapsto (0, 5)$. Where does $(3, 1)$ go?
Answer
$(6, 5)$. By superposition, $T(3, 1) = 3\,T(\mathbf{e}_1) + 1\,T(\mathbf{e}_2) = 3(2,0) + (0,5) = (6, 0) + (0, 5) = (6, 5)$.Q9. Why does removing all the nonlinear activation functions make a deep neural network no more powerful than a single layer?
Answer
Because the **composition of linear maps is itself a linear map** (Chapter 8). A stack of matrices with nothing between them multiplies out to one equivalent matrix, so depth buys nothing. The nonlinearities are what give deep networks their expressive power.Q10. Which of the following is not a place linear algebra plays a central role? (a) ranking web pages; (b) compressing an image; (c) the transcendence of $\pi$; (d) transforming 3D models for a game; (e) describing a qubit's state.
Answer
**(c).** The transcendence of $\pi$ is a result in number theory/analysis with no linear-algebra content. The other four are all squarely linear algebra (PageRank, SVD compression, graphics transforms, quantum state vectors).Q11. The chapter's central one-sentence idea is "a matrix is a ___ that transforms space." Fill in the blank, and name the recurring tool that makes this visible.
Answer
A matrix is a **function** that transforms space. The recurring tool is the **2D transformation visualizer** (`visualize_2d`), which shows what any $2\times 2$ matrix does to the unit square — introduced in this chapter and reused in nearly every chapter after.Q12. Compound interest is given as the chapter's headline example of a nonlinear process. In one sentence, why is it nonlinear?