Some Math Connotations Demystified 数学内涵解密

This Taiwanese Math Prof is very approachable in clarifying the doubts in an unconventional way different from the arcane textbook definitions. Below are his few key tips to breakthrough the “mystified”concepts :

1. “Dual Space“(对偶空间) : it is the evaluation  of a “Vector Space”. 

Example: A student studies few subjects {Math, Physics, English, Chemistry…}, these subjects form a “Subject Vector Space” (V), if we associate the subjects with weightages  (加权) , say, Math 4, Physics 3, English 2, Chemistry 1, the “Weightage Dual Space” of V will be W= {4, 3, 2, 1}.

2. Vector: beyond the meaning of a physical vector with direction and value, it extends to any “object” which can be manipulated (eg. cancel off 抵消 / multiple n times, …) by the 4 operations “+, – , x, / ” in a Field F = {R or Z2 =(0,1) or Zp for any prime p, …}. 

Eg. \alpha_{1}.v_{1} + \alpha_{2}.v_{2} + \alpha_{3}.v_{3}, \forall \alpha_{j} \in F

3. “Linear Algebra” (represented in Matrix) : its core motive is to find the best “angle” (basis) from which to view the (linear) transformation – ie mathematically find the Eigen- (特征) Vectors and Eigen-Values.  

Application in Google Search : find the webpages with best page rankings.

4. Algebraic Topology and Algebraic Geometry are the 2 toughest Math subjects, both use extensively (Modern) Algebra as the tool.

To know a Math topic really well means “you can explain it to any person on the street and make him understand” –(David Hilbert).

5. Most High School Math teachers are good at solving math problems, but few know the intrinsic connotations (内涵) of Math ideas. The later is important to motivate students’ passion in Math.

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