$$

Tasks

本學期懸賞已截止

The following problems are either without answer or can be improved. Everyone is encouraged to work on them. You will received extra points as a reward once if you finish a problem correctly.

• 101: 6(a), 6©, 7(a), 7(b)
• 102: 4
• 103: 4(b), 4©, 4(d), 5(a), 5(b)
• 104: 3
• 105: 4(d), 4(e)
• 106: 2(b), 3, 4, 5
• 107: 2
• 108: 3, 4, 5, 6
• 109: 3 to end
• 110: 2, 3©, 3(d)
• 111: all
• 112: 1, 2, 3, 4(a~c), 5
• 113: all
• 114: all
• 201: 3(b), 4, 5, 6(b)
• 202: 2(b), 2©
• 203: 4, 5(a), 6, 7
• 204: 6
• 205: 4, 5
• 206: 3, 4, 5, 6
• 207: 4(b), 4©, 4(d), 5(a), 5(b)
• 208: 1©, 4 to end
• 209: all
• 210: all
• 211: all
• 212: all
• 213: all
• 214: all
• 215: all
• 301: 4(a~d)
• 302: 5(b), 6
• 303: 3, 4, 5
• 304: 1(d), 2(d~f), 3(b)
• 305: 4(a~d), 5(b)
• 306: 4
• 307: 2d, 3, 4
• 308: all
• 309: 2 ~ end
• 310: all
• 311: all
• 312: all
• 313: all
• 314: all
• 315: all

Once the group has submitted the answers, the note will be locked. This means you can no longer edit, but you are able to see the source code. If you wish to work on a particular problem:

1. Go to HackMD and open a new note by clicking "+ New note".
2. Add the following line at the top of your note. Then copy the source code of the problem to your note.

{%hackmd 5xqeIJ7VRCGBfLtfMi0_IQ %}

Thus, you may use some macros, such as

$\mathbf{v}$.

3. Once you are done, copy the link of your note and send it to Jephian Lin through Discord. If it is correct, then you will receive the points. (Only the first one will receive the points, but answers with different insights are welcome.)

Finished Tasks:

• 101: 2(鄭宗祐 +2), 5(b)(鄭宗祐), 6(a)(鄭宗祐), 6©(鄭宗祐), 7©(許豐有)
• 102: 3(a~d)(許豐有 +2), 3(e~l)(鄭宗祐 +4)
• 103: 3(b,c)(鄭宗祐 +1)
• 105: 4(a)(蔡睿丞)
• 107: 3©(朱曼華), 3(d)(朱曼華)
• 109: 1(a~d)(張永賦 +2), 2(a,b)(張永賦)
• 110: 1(張永賦), 3(a,b)(鄭宗祐 +1)
• 112: 4(d)(鄭宗祐)
• 206: 2(b)(鄭宗祐)
• 208: 1(a,b)(張永賦 +1), 2(a~c)(張永賦 +1.5), 3(a~c)(張永賦 +1.5)
• 302: 2(d~f)(吳愷杰 +1.5), 5(a)(孫心)
• 303: 2(e)(孫心)
• 309: 1(張永賦 +2)

You will receive 0.5 points for each problem.

Math Runway Exercise by November 30. You will get 1 extra point if you finished one exercise correctly.

鄭宗祐, 湯昆璋, 黃經幃

Linear Algebra I –- Collaborative notes, 2022FMath103A

1. Linear geometry

1. Vector, length, and angle
2. Subspaces in Rn
3. Column space of a matrix
4. Row space of a matrix
5. Projection and reflection
6. Affine subspaces in Rn
7. Solution set of Ax = b
8. Row operations
9. Finding a particular solution
10. Finding the homogeneous solutions
11. Number of solutions
12. Matrix inverse
13. Elementary matrices
14. Four fundamental subspaces

a. Sage: Matrices and linear equations

Basic geometry & subspaces
101 --> 102 --> {103, 104} --> 105

Affine subspace & solutions
(102 -->)
106 --> 107 --> 108 --> {109, 110} --> 111

Topics
112, 113, 114, 10a

2. Linear spaces

1. Linear independence
2. Basis
3. Column space, left kernel, and their bases
4. Row space, kernel, and their bases
5. Basis exchange lemma
6. Dimension, expanding and shrinking lemmas
7. Rank and nullity
8. Vector space
9. Subspaces in a vector space
10. Common vector spaces
11. Constructing new subspaces
12. Constructing new vector spaces
13. Orthogonal geometry
14. Gram–Schmidt orthogonalization
15. Direct sum of orthogonal subspaces

Spaces in Rn
201 --> 202 --> {203, 204} --> 205 --> 206 --> 207

Abstract spaces
208 --> 209 --> 210

Operations of spaces
211 -->  212

Inner product space
213 --> 214 --> 215

3. Linear functions

1. Function basics
2. Linear function
3. Matrix as a linear function
4. Linear function as a matrix
5. Vector representation in Rn
6. Vector representation in a vector space
7. Change of basis
8. Isomorphism
9. Matrix representation in Rn
10. Matrix representation in a vector space
11. Lagrange polynomials and Vandermonde matrix
12. Sylvester matrix and resultant
13. Understanding the spectral decomposition
14. Understanding the singular value decomposition
15. Understanding the Jordan canonical form

Linear function
301 --> 302 --> 303 -->304

Vector and matrix representations
305 --> 306 --> 307 --> 308 --> 309 --> 310

Topics
311, 312, 313, 314, 315