# Week 4 Quiz Answers: Linear Systems & Documents

## Situation 1: Vehicle Matrix D and Matrix A

### Question 1: **TRUE**
**Statement:** The null space vector provided is a solution to the system $Dx = 0$, where $D$ is the full matrix of vehicles, including the value_score column, and the i-th value in $x$ corresponds to the i-th column of $D$.

**Explanation:**

By definition, a null-space vector $x$ satisfies $Dx = 0$. In the notebook, the vector shown as a "basis for the null space" of $D$ is exactly such a vector, with one entry per column of $D$.

**Answer: TRUE**

---

### Question 2: **FALSE**
**Statement:** The matrix $D$ has a rank less than its number of columns. Therefore, if we pick a vector $b$ with a compatible number of rows, the system of equations $Dx = b$ has no solution.

**Explanation:**

Rank less than the number of columns means the columns of $D$ are linearly dependent, so the null space is nontrivial. This implies that solutions (when they exist) are **not unique**, but it does not mean there is never a solution.

Key insight:
- Any $b$ in the **column space** of $D$ gives at least one solution
- Only $b$ **outside** the column space gives no solution

The statement incorrectly claims that ANY vector $b$ has no solution, which is false.

**Answer: FALSE**

---

### Question 3: **FALSE**
**Statement:** Given $A$ as described in the notebook, it is possible that the column weight is a perfect linear combination of the rest of the columns in $A$. I.e., we can find $a, b, c$ such that the following is true:

$$a \cdot \text{acc} + b \cdot \text{model_year} + c \cdot \text{cyl} = \text{weight}$$

**Explanation:**

In the notebook, the matrix formed from the columns (weight, acceleration, model_year, cylinders) is described as having an **empty null space**.

Empty null space means:
- Only the zero vector is in the null space
- The matrix has **full column rank**
- There is no nontrivial linear dependence among the columns

Therefore, weight **cannot** be an exact linear combination of acceleration, model_year, and cylinders.

**Answer: FALSE**

---

### Question 4: **TRUE**
**Statement:** $A'$ is invertible. Therefore, there is only one way to add multiples of the columns of $A'$ to "create" $b'$. That is, there is only one linear combination of the columns of $A'$ that yields $b'$.

**Explanation:**

If $A'$ is invertible, the system $A'x = b'$ has a **unique solution**.

Interpreting this geometrically:
- The entries of $x$ are coefficients in a linear combination of the columns of $A'$
- There is exactly one coefficient vector $x$
- Hence exactly one linear combination of the columns that produces $b'$

This is a fundamental property of invertible matrices: the transformation is one-to-one and onto.

**Answer: TRUE**

---

## Situation 2: Document Matrix C and Cosine Distance

### Question 5: **TRUE**
**Statement:** Consider the document $d$ we created. Document $d$ is in the row space of the matrix $C$.

**Explanation:**

The document $d$ is defined by adding existing document rows (for example, $d = \text{row}_1 + \text{row}_2$).

The **row space** of $C$ consists of all linear combinations of its rows. Since $d$ is such a combination, it lies in the row space of $C$.

**Mathematical formulation:**
$$d = c_1 \cdot \text{row}_1 + c_2 \cdot \text{row}_2 + \cdots + c_n \cdot \text{row}_n$$

This is exactly the definition of membership in the row space.

**Answer: TRUE**

---

### Question 6: **FALSE**
**Statement:** The heatmap visualization of cosine distances allows us to observe the angle between document vectors.

**Explanation:**

The heatmap shows **cosine distances**, typically defined as:
$$\text{cosine distance} = 1 - \cos(\theta)$$

This reflects similarity indirectly, but it does **not** display the actual angle $\theta$ itself.

What you can observe:
- Which pairs are more or less similar
- Relative similarity relationships

What you **cannot** observe:
- The actual angle values $\theta$ directly

To see angles, you would need to compute $\theta = \arccos(\text{similarity})$.

**Answer: FALSE**

---

### Question 7: **TRUE**
**Statement:** By looking at the cosine distance matrix (at the bottom of the notebook), we see evidence of documents that do not share any terms in common.

**Explanation:**

The document vectors here are nonnegative term-frequency vectors.

If two documents share **no terms** in common:
- Every product of corresponding entries is zero
- Their dot product is zero: $\vec{d}_1 \cdot \vec{d}_2 = 0$
- Cosine similarity $= \frac{0}{\|\vec{d}_1\| \|\vec{d}_2\|} = 0$
- Cosine distance $= 1 - 0 = 1$

Therefore, **off-diagonal entries equal to 1** in the cosine distance matrix are evidence of document pairs that have no overlapping terms (in the chosen vocabulary).

**Answer: TRUE**

---

### Question 8: **FALSE**
**Statement:** In the cosine distance matrix, if there were 0 values on the off-diagonal, that would indicate two documents that must be identical, i.e., exactly the same.

**Explanation:**

Cosine distance $= 0$ means cosine similarity $= 1$, which means:
$$\cos(\theta) = 1 \implies \theta = 0°$$

This means the vectors are **parallel**: one is a positive scalar multiple of the other.

$$\vec{d}_1 = k \cdot \vec{d}_2 \text{ for some } k > 0$$

For term-frequency vectors, this means:
- Same **relative term pattern** (proportions)
- Possibly different **overall counts** (magnitudes)

**Example:**
- Document 1: "cat cat dog" → $[2, 1]$
- Document 2: "cat cat cat dog dog dog" → $[3, 3] = 1.5 \times [2, 2]$... wait, let me recalculate
- Document 2: "cat cat dog" repeated twice → $[4, 2] = 2 \times [2, 1]$

These have cosine similarity = 1 but are not identical entry-by-entry.

The documents need **not** be exactly the same.

**Answer: FALSE**

---

## Answer Summary

| Question | Situation | Answer |
|----------|-----------|--------|
| Q1 | Vehicle Matrix (Situation 1) | **TRUE** |
| Q2 | Vehicle Matrix (Situation 1) | **FALSE** |
| Q3 | Vehicle Matrix (Situation 1) | **FALSE** |
| Q4 | Vehicle Matrix (Situation 1) | **TRUE** |
| Q5 | Document Matrix (Situation 2) | **TRUE** |
| Q6 | Document Matrix (Situation 2) | **FALSE** |
| Q7 | Document Matrix (Situation 2) | **TRUE** |
| Q8 | Document Matrix (Situation 2) | **FALSE** |