Chapter 7 Quiz — Matrices as Functions
Twelve conceptual checks. Try each before opening the answer. These test understanding, not arithmetic speed — if you can answer them, you have the chapter.
Q1. What is the single best one-sentence answer to "what does a matrix do to a vector?"
Answer
It applies a linear transformation to the vector — it moves the vector by rotating, scaling, shearing, reflecting, or projecting space. A matrix is a transformation written down. *(The whole chapter in one line.)*Q2. The first column of a $2\times 2$ matrix $A$ is $(3, -1)$. Without any other information, what is $A\mathbf{e}_1$?
Answer
$A\mathbf{e}_1 = (3, -1)$. The first column *is* the image of $\mathbf{e}_1$ — that is the definition of the matrix of a transformation. Multiplying by a basis vector extracts the corresponding column.Q3. Why is the matrix-vector product equal to the weighted sum of the columns, and where does that fact come from?
Answer
Because any vector is $\mathbf{v} = x\mathbf{e}_1 + y\mathbf{e}_2$, and linearity (superposition) gives $A\mathbf{v} = xA\mathbf{e}_1 + yA\mathbf{e}_2 = x(\text{col }1) + y(\text{col }2)$. It is *derived* from linearity, not a rule to memorize. The weights are the entries of $\mathbf{v}$.Q4. Is "rotate the plane by $45°$, then slide everything one unit to the right" a linear transformation? Why or why not?
Answer
No. The slide (translation) moves the origin to $(1,0)$, and a linear transformation must fix the origin ($T(\mathbf{0})=\mathbf{0}$). The combined map is *affine*, not linear. No $2\times 2$ matrix can perform it; computer graphics handles translation with homogeneous coordinates (Chapter 12).Q5. Derive (don't recite) where $\mathbf{e}_1$ and $\mathbf{e}_2$ go under a counterclockwise rotation by $\theta$, and assemble the rotation matrix.
Answer
$\mathbf{e}_1=(1,0)$ swings to the unit-circle point at angle $\theta$: $(\cos\theta, \sin\theta)$. $\mathbf{e}_2=(0,1)$ is east turned $90°$, so its image is east's image turned another $90°$ left: $(-\sin\theta, \cos\theta)$. As columns: $R(\theta)=\begin{bmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{bmatrix}$.Q6. What does a determinant of $0$ tell you about a transformation, and why can such a transformation not be undone?
Answer
It means space was flattened onto a lower dimension (e.g. the plane squashed onto a line) — the matrix is *singular*. It cannot be undone because it destroys information: many input points map to the same output (the whole vertical direction collapsed to nothing), so there is no rule to recover the input. *(Invertibility: Chapter 9.)*Q7. A negative determinant signals what, geometrically?
Answer
Orientation reversal — the transformation includes a *flip* (a reflection), turning a counterclockwise loop into a clockwise one, or a left hand into a right hand. You cannot achieve it by rotating the page; you must mirror it.Q8. Why does the product $A\mathbf{v}$ always land inside the span of the columns of $A$?
Answer
Because $A\mathbf{v}$ *is* a linear combination of the columns (the weighted-sum-of-columns formula). Anything built as a combination of the columns lies in their span by definition. This is the seed of the **column space** $C(A)$, Chapter 13.Q9. Reflection across the line $y = x$ has matrix $\begin{bmatrix}0&1\\1&0\end{bmatrix}$, which has no negative entries. How can it be a reflection? How would you confirm it's orientation-reversing?
Answer
Read the columns: $\mathbf{e}_1\to(0,1)$ and $\mathbf{e}_2\to(1,0)$ — east and north *swap*, which is exactly mirroring across $y=x$. Orientation is judged by the *sign of the determinant*, not by entry signs: $\det = 0\cdot0 - 1\cdot1 = -1 < 0$, so it flips orientation. Entry signs are not a reliable orientation test.Q10. What is the transpose of a matrix, and is $A^{\mathsf{T}}$ generally the same transformation as $A$?
Answer
The transpose flips rows and columns: $(A^{\mathsf{T}})_{ij}=a_{ji}$. It is generally a *different* transformation (e.g. a horizontal shear's transpose is a vertical shear). It is *not* the inverse. It coincides with $A$ only for *symmetric* matrices, where $A^{\mathsf{T}}=A$.Q11. You combine two transformations: first $B$, then $A$. How do you find the columns of the combined transformation's matrix, using only this chapter's tools?
Answer
Track each basis vector through both steps: $\mathbf{e}_1 \to B\mathbf{e}_1 \to A(B\mathbf{e}_1)$, and likewise for $\mathbf{e}_2$. The final landing spots $A(B\mathbf{e}_1)$ and $A(B\mathbf{e}_2)$ are the columns of the combined matrix. Chapter 8 will call this combined matrix the product $AB$ (with $A$ on the left because it acts last).Q12. Why does this book insist on the "columns are images of basis vectors" picture instead of teaching matrix multiplication as a row-times-column rule first?