Capstone & Appendix · Study deck
IoT Mathematics: Linear Algebra and Probability
Several sensor values form a vector, while noise makes each value unsure.
Test Tessa is your guide for this deck.
After studying this chapter
Learning objectives
You will be able to:
- Explain: Likewise, an observer may see public values such as 8 and 19, while recovering the private exponents requires solving the discrete logarithm problem for properly chosen large parameters.
- Explain: Diffie-Hellman relies on forward calculations such as 5^7 mod 23 = 17 being easy while the reverse problem is hard.
- Explain: The eavesdropper sees the public starting colour and both exchanged mixtures but cannot reverse them to recover either private colour.
- Explain: The analogy explains one-way computation, not authentication; a real exchange still needs authenticated peers and safe key handling.
Major section
The Core Idea
Modular arithmetic is "clock arithmetic." After reaching a maximum value, numbers wrap around to zero.
- Notation: $a \mod n$ = remainder when a is divided by n.
- Key Property: Easy to compute forward, hard to reverse.
- Two IoT devices can agree on a shared secret over an insecure channel.
Major section
The Core Idea (continued)
This "trapdoor" property enables public-key cryptography.
- The eavesdropper sees the public starting colour and both exchanged mixtures but cannot reverse them to recover either private colour.
- Likewise, an observer may see public values such as 8 and 19, while recovering the private exponents requires solving the discrete logarithm problem for properly chosen large parameters.
- The analogy explains one-way computation, not authentication; a real exchange still needs authenticated peers and safe key handling.
Major section
The Core Idea
Entropy measures information content or uncertainty.
- More unpredictable data requires more bits to represent.
- Where $p_i$ is the probability of each outcome.
- The alarm provides only 0.08 bits of information per reading (very predictable).
- Key Insight: To double capacity, you need to roughly quadruple SNR.
Major section
Checkpoint: Security and Signals
That wraparound supports the forward operations used in the key-exchange example.
- Diffie-Hellman relies on forward calculations such as 5^7 mod 23 = 17 being easy while the reverse problem is hard.
- Authentication and key handling remain separate requirements.
- Complex numbers track amplitude and phase, while information theory connects entropy, SNR, bandwidth, and Shannon capacity.
Deck summary
Key takeaways
Modular arithmetic is "clock arithmetic." After reaching a maximum value, numbers wrap around to zero.
- This "trapdoor" property enables public-key cryptography.
- Entropy measures information content or uncertainty.
- That wraparound supports the forward operations used in the key-exchange example.
Retrieval practice
Recall check 1 of 2
Test Tessa says: answer from memory, then check your reasoning.
Q1An accelerometer produces an ordered set of axis readings. What does the vector magnitude describe?
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Answer: A The chapter uses the square root of summed squared components to describe total magnitude.
Q2A sensor-fusion design stores a covariance matrix. What do its off-diagonal entries describe?
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Answer: D The chapter distinguishes cross-variable correlation from diagonal variance.
Retrieval practice
Recall check 2 of 2
Test Tessa says: answer from memory, then check your reasoning.
Q3Place each mathematical responsibility where it lives so you can trace a decision back through network and signal calculations to bounded device measurements.
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Answer: A Mathematical reasoning is an evidence chain: measure with known limits, calculate whether the path preserves what matters, then make a bounded model-based decision.
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Answers
Answer key.
- A · The chapter uses the square root of summed squared components to describe total magnitude.
- D · The chapter distinguishes cross-variable correlation from diagonal variance.
- A · Mathematical reasoning is an evidence chain: measure with known limits, calculate whether the path preserves what matters, then make a bounded model-based decision.