Do It By Hand | Eriva Labs

Part 1: The Underlying Mechanics

Problem 1: Vector Construction (Position is Identity) A credit-risk model locks its feature order as [Credit Score (in 100s), Years Employed, Debt Ratio]. An applicant has been employed for 4 years, carries a Debt Ratio of 0.3, and has a Credit Score of 720 (encode as $7.2$ ). Construct the correctly-ordered feature vector $x$ .

Reveal Solution

$x = \begin{bmatrix} 7.2 \\ 4 \\ 0.3 \end{bmatrix}$

The applicant's facts were given out of order (Years, Debt Ratio, Credit Score) — the vector must still follow the locked order (Credit Score, Years, Debt Ratio), not the order the facts happened to be mentioned in.

Problem 2: Position Swap Consequence A weather model locks its order as [Temperature (°F), Humidity (%), Wind Speed (mph)]. Monday's true weather is correctly logged as $[70, 60, 12]$ . An intern re-logs the exact same physical weather for Tuesday but accidentally swaps positions 1 and 2, recording $[60, 70, 12]$ . According to the locked feature order — not what the intern intended — what does the vector $[60, 70, 12]$ actually claim Tuesday's Temperature and Humidity were?

Reveal Solution

Position 1 is always read as Temperature and position 2 as Humidity, regardless of intent. So $[60, 70, 12]$ claims Temperature $= 60°F$ and Humidity $= 70\%$ — a physically different (and wrong) reading from Monday's actual $70°F$ / $60\%$ . The math has no way to know positions were swapped; it just reads position 1 as Temperature, every time. This is exactly why the order must never be scrambled.

Problem 3: Linear Combination Calculate $2v_1 - v_2$ where $v_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$ and $v_2 = \begin{bmatrix} 4 \\ 2 \end{bmatrix}$ .

Reveal Solution

$2 \begin{bmatrix} 1 \\ 0 \end{bmatrix} - \begin{bmatrix} 4 \\ 2 \end{bmatrix} = \begin{bmatrix} 2 \\ 0 \end{bmatrix} - \begin{bmatrix} 4 \\ 2 \end{bmatrix} = \begin{bmatrix} -2 \\ -2 \end{bmatrix}$

This is scalar multiplication and vector subtraction composed: scale $v_1$ by $2$ , then subtract $v_2$ .

Problem 4: 3D Dot Product (Mixed Weights) Calculate the dot product $w \cdot x$ where $w = \begin{bmatrix} 2.0 \\ -1.5 \\ 0.5 \end{bmatrix}$ and $x = \begin{bmatrix} 3.0 \\ 2.0 \\ 4.0 \end{bmatrix}$ .

Reveal Solution

$w \cdot x = (2.0)(3.0) + (-1.5)(2.0) + (0.5)(4.0) = 6.0 - 3.0 + 2.0 = \mathbf{5.0}$

Problem 5: The "Zero Weight" Impact Calculate $w \cdot x$ where $w = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ and $x = \begin{bmatrix} 100 \\ 500 \\ 999 \end{bmatrix}$ .

Reveal Solution

$w \cdot x = (0)(100) + (0)(500) + (0)(999) = 0 + 0 + 0 = \mathbf{0}$

No matter how large a feature's measurement is, a weight of exactly $0$ erases its influence on the score completely.

Part 2: Applied Scenario: The VC Greenlight Decision

Imagine you are building an AI decision model for a Venture Capital firm. The firm receives hundreds of startup pitches each week and wants to predict which startup pitch to greenlight based on its Unicorn Potential (whether a startup has the potential to become a billion-dollar company or fail).

First, the algorithm loads the firm's Unicorn Potential Weights ( $w$ ) across four locked criteria: [Team Experience, Market Size, Competition, Risk]. This represents what the VC firm believes indicates a massive outcome, which directly influences whether a pitch gets greenlit or passed:

w = \begin{bmatrix} 4.0 \\ 2.0 \\ -3.0 \\ 1.0 \end{bmatrix} \begin{matrix} \text{(Team Experience Weight)} \\ \text{(Market Size Weight)} \\ \text{(Competition Weight)} \\ \text{(Risk Weight)} \end{matrix}

The partners heavily value a proven Team ( $+4.0$ ) and a massive Market ( $+2.0$ ). They strongly penalize crowded Competition ( $-3.0$ ), and maintain a mild appetite for high-upside Risk ( $+1.0$ ).

Now, two startups submit their pitch decks:

OmniFlow (The Hyped Veteran Pitch): Team Experience $1.0$ (repeat founders), Market Size $0.3$ (small enterprise niche), Competition $1.0$ (crowded market), Risk $0.2$ (safe, conventional product).
Solaris AI (The Scrappy Moonshot): Team Experience $0.3$ (first-time founders), Market Size $1.0$ (massive untapped market), Competition $0.0$ (no direct rivals), Risk $0.8$ (high technical risk).

Problem 6: Two Pitches, One Decision

Part A: Encode both pitches into feature vectors, $x_{\text{OmniFlow}}$ and $x_{\text{Solaris}}$ , respecting the locked position order above.

Part B: Calculate the raw Dot Product score ( $w \cdot x$ ) for each startup.

Part C: In one sentence, which startup does the firm's Unicorn Potential model favor, and which single feature is doing the most damage to the weaker pitch?

Reveal Solution

Part A: $x_{\text{OmniFlow}} = \begin{bmatrix} 1.0 \\ 0.3 \\ 1.0 \\ 0.2 \end{bmatrix} \qquad x_{\text{Solaris}} = \begin{bmatrix} 0.3 \\ 1.0 \\ 0.0 \\ 0.8 \end{bmatrix}$

Part B: $w \cdot x_{\text{OmniFlow}} = (4.0)(1.0) + (2.0)(0.3) + (-3.0)(1.0) + (1.0)(0.2) = 4.0 + 0.6 - 3.0 + 0.2 = \mathbf{1.8}$ $w \cdot x_{\text{Solaris}} = (4.0)(0.3) + (2.0)(1.0) + (-3.0)(0.0) + (1.0)(0.8) = 1.2 + 2.0 + 0.0 + 0.8 = \mathbf{4.0}$

To visually verify how each term contributes to the final scores, examine the computational dataflow below:

VC Greenlight Decision Flow Diagram — Comparative dataflow for the VC Greenlight model: Although OmniFlow scores high on Team Experience ($+4.0$), the intense Competition penalty ($-3.0$) pulls its raw score down to $+1.8$. Solaris AI wins the greenlight ($+4.0$) thanks to zero competition and a massive market.

Part C: The Unicorn Potential model favors the scrappy moonshot Solaris AI ( $4.0$ vs. $1.8$ ). While human investors might be dazzled by OmniFlow's repeat-founder team ( $+4.0$ ), the model exposes that intense Competition ( $1.0 \times -3.0 = -3.0$ ) drastically drags down its potential to return venture-scale returns.

Just like the Netflix scores of $+9.5$ and $-1.5$ , these are raw linear scores, not final funding decisions — they still need a Bias and an Activation Function (Topic 2) before they become clean, comparable probabilities.