lecture_EE1G1_lec1_math_echelon_2025-09-01

📌 Overview

The Integrated Math component introduces mathematical instruments crucial for interpreting and modeling linear circuits and for broader electrical engineering disciplines. It combines intuitive definitions with hands-on practice, synchronizing with taught circuit topics.

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🎯Learning Objectives

• understand online systems ✅ 2025-09-15

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📋 Requirements

link

Overall Assessment: Grasple Exercises

To pass the Integrated Math component, you must meet the following by Sunday 9 November 2025, 23:59h:

• Obtain a green check mark for each of the 8 Basic Grasple exercise sets.
• Obtain at least a yellow/orange check mark for each of the 8 Advanced Grasple exercise sets.

Check Mark Criteria:

• Green: Each question answered and ≥80% correct on the first attempt.
• Yellow/Orange: Each question answered and ≥50% correct on the first attempt.
• Note: Unlimited attempts allowed; exercises vary each time. Your best attempt counts. Grasple tutorial and “Optional” sets do not count towards these requirements.

Repair Option (if initial requirements not met):

• Condition: If you fail the Integrated Math component but obtained at least 10 yellow/orange check marks (total from basic and/or advanced sets) by 2025-11-09 23:59.
• Process: You will gain access to 3 new Grasple exercise sets in Quarter 2.
• Requirement: Obtain a green check mark for all 3 of these new sets by the end of Quarter 2 to pass the component.

Assignment	Link	Started	Done
Grasple	link

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submission

• Due Date: 2025-11-09 (for initial Grasple requirements)
• Submission Method: Grasple exercises (online)

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notes

Education Method: Follows a Preparation (pre-lecture video) – Participation (lecture, exercises, practical applications) – Post-lecture work (homework, Grasple exercises) cycle.

Important: Start completing Grasple exercises early! They can be a lot of work and may require multiple attempts. Keeping up will help you stay focused on other exams and reinforce your learning.

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✍️ Notes

Stelsels oplossen

Gauss-algoritme (Gauss-eliminatie)

Het Gauss-algoritme is een systematische methode om een matrix om te zetten in Echelon Form (rij-echelonvorm) door middel van elementaire rij-operaties. Dit wordt vaak gebruikt om stelsels van lineaire vergelijkingen op te lossen.

Doel: Transformeer de matrix zodat deze voldoet aan de criteria van de Echelon Form.

Elementaire Rij-Operaties:

1. Rijen verwisselen: Wissel twee rijen van plaats.
   • $R_i\leftrightarrow R_j$
   • Voorbeeld: $$\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\stackrel{R_1\leftrightarrow R_2}{\to }\left[\begin{array}{cc}3& 4\\ 1& 2\end{array}\right]$$
2. Rij vermenigvuldigen: Vermenigvuldig een rij met een niet-nul scalair ($k\ne 0$).
   • $k\cdot R_i\to R_i$
   • Voorbeeld: $$\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\stackrel{2R_1\to R_1}{\to }\left[\begin{array}{cc}2& 4\\ 3& 4\end{array}\right]$$
3. Rij optellen bij een andere rij: Tel een veelvoud van één rij op bij een andere rij.
   • $R_i+k\cdot R_j\to R_i$
   • Voorbeeld: $$\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\stackrel{R_2-3R_1\to R_2}{\to }\left[\begin{array}{cc}1& 2\\ 3-3(1)& 4-3(2)\end{array}\right]=\left[\begin{array}{cc}1& 2\\ 0& -2\end{array}\right]$$

Stappen (algemeen):

1. Begin bij de meest linkse kolom die niet volledig uit nullen bestaat.
2. Gebruik rij-verwisselingen om een niet-nul getal (pivot) naar de bovenste positie in die kolom te brengen.
3. Gebruik rij-operatie 3 om alle getallen onder deze pivot nul te maken.
4. Ga naar de volgende rij en de volgende kolom naar rechts, en herhaal het proces voor de submatrix die overblijft, totdat de Echelon Form is bereikt.

Voorbeeld (naar Echelon Form):

$$\left[\begin{array}{cccc}0& 1& 2& 3\\ 1& 1& 0& 1\\ 2& 4& 4& 8\end{array}\right]$$

1. $R_1\leftrightarrow R_2$ (breng een niet-nul naar de leidende positie) $$\left[\begin{array}{cccc}1& 1& 0& 1\\ 0& 1& 2& 3\\ 2& 4& 4& 8\end{array}\right]$$
2. $R_3-2R_1\to R_3$ (maak getallen onder de leidende 1 nul) $$\left[\begin{array}{cccc}1& 1& 0& 1\\ 0& 1& 2& 3\\ 0& 2& 4& 6\end{array}\right]$$
3. $R_3-2R_2\to R_3$ (maak getallen onder de leidende 1 van de tweede rij nul) $$\left[\begin{array}{cccc}1& 1& 0& 1\\ 0& 1& 2& 3\\ 0& 0& 0& 0\end{array}\right]$$ De matrix is nu in Echelon Form.

📝 Summary

final result of gauss’s algorithm

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💡 Explanation

Echelon Form (Row Echelon Form - REF)

A matrix is in Echelon Form if:

1. All zero rows are at the bottom. $$\left[\begin{array}{ccc}\cdot & \cdot & \cdot \\ 0& \cdot & \cdot \\ 0& 0& 0\end{array}\right]$$
2. The leading entry (first non-zero entry) of each row is to the right of the leading entry of the row above it. $$\left[\begin{array}{ccc}1& \cdot & \cdot \\ 0& 1& \cdot \\ 0& 0& 0\end{array}\right]\text{or}\left[\begin{array}{ccc}2& \cdot & \cdot \\ 0& 5& \cdot \\ 0& 0& 0\end{array}\right]$$
3. All entries in a column below a leading entry are zero. $$\left[\begin{array}{ccc}1& \cdot & \cdot \\ 0& 1& \cdot \\ 0& 0& 0\end{array}\right]$$ Example: $$\left[\begin{array}{ccccc}1& 0& 3& -1& 0\\ 0& 1& 1& -1& 0\\ 0& 0& 0& 0& 0\end{array}\right]$$

Reduced Row Echelon Form (RREF)

A matrix is in RREF if it is in Echelon Form and additionally: 4. Each leading entry is 1. (Called a “leading 1”). $\left[\begin{array}{ccc}1& \cdot & \cdot \\ 0& 1& \cdot \\ 0& 0& 0\end{array}\right]$ 5. Each leading 1 is the only non-zero entry in its column. $\left[\begin{array}{ccc}1& 0& \cdot \\ 0& 1& \cdot \\ 0& 0& 0\end{array}\right]$ Example: $\left[\begin{array}{ccccc}1& 0& 3& -1& 0\\ 0& 1& 1& -1& 0\\ 0& 0& 0& 0& 0\end{array}\right]$

🖼️ Diagrams & Visuals

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📚 Related Resources

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🔗 Resources

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❓ Post lecture

$$\begin{array}{cccc}1& -4& -2& -13\\ 0& 5& -2& 22\\ 0& -15& 8& -68\end{array}$$

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📖 Homework