$$

Tasks

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.

The last day to submit your answers is the day of final exam.

• 406: 3, 4 all, 5(b)
• 407: 6
• 412: 3(e), 4, 5
• 503: 2c, 2d, 5
• 504: 3, 4, 5
• 505: 7
• 506: 5, 6, 7, 8, 9, 10
• 507: 4, 5, 6, 7, 8, 9
• 602: 5, 6, 7
• 603: 3, 6, 7
• 604: 3b, 4, 5, 6, 7
• 605: 3c, 3d, 6, 7, 8
• 606: 2, 3, 4, 5, 6
• 607: 3c, 4, 5

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:

• 401: 3 ()
• 402: 1(b) (鄭宗祐), 2a (), 3b (), 4 (), 5 ()
• 403: 6 ()
• 404: 3 (鄭宗祐)
• 405: 3(b) (), 5(a,b) ( +1), 6(a,b,c) ( +1.5)
• 406: 4(a,b,c) ( +1.5)
• 407: 7(a,b) (蔡睿丞 +1)
• 411: 1 (張永賦 +1.5), 3 (張永賦 +1.5)
• 412: 1(a,b,c) ( +1.5), 2 ( +1), 3(a,b,c,d) ( +2)
• 501: 5(b~e) (孫心 +2), 6 (孫心 +1)
• 504: 3(a,b) (孫心 +1), 4(a,b,c,d) (孫心 +2), 5 (孫心)
• 506: 3(a,b,c,d) (孫心 +2), 4(b,c) (孫心 +2)
• 603: 2c (孫心), 4 (蔡睿丞, 孫心), 5 (孫心)
• 604: 2c (蔡睿丞)
• 605: 4 (孫心 +1), 5 (孫心)
• 608: 2 (蔡睿丞 +1.5)

You will receive 0.5 points for each problem.

Linear Algebra II –- Collaborative notes, 2023SMath104A

Linear Algebra I –- Collaborative notes, 2022FMath103A

4. Determinant

1. Determinant for small matrices
2. Invertible matrix as the product of elementary matrices
3. Elementary matrix acts on columns
4. Elementary matrix acts on rows
5. Definition of the determinant
6. Invertibility
7. Matrix multiplication and transpose
8. Block matrix
9. Distributive law and Laplace expansion
10. Adjugate
11. Cramer's rule
12. Permutation matrix
13. Permutation expansion
14. Interpretation through graph theory
15. Determinant is well-defined

Determinant for small matrices
401

Geometric interpretations
402 --> {403, 404}

Definition and properties
405 --> {406, 407, 408} --> 409

Adjugate
410 --> 411

Permutation expansion
412 --> 413 --> {414, 415}

5. Diagonalization

1. Find a good basis
2. Quadratic curve
3. Recurrence relation
4. Linear differential equation
5. Matrix exponential
6. Characteristic polynomial
7. Coefficients of the characteristic polynomial
8. Diagonalization
9. Algebraic multiplicity and geometric multiplicity
10. Eigenspace
11. Graph and characteristic polynomial
12. Cayley–Hamilton theorem
13. Minimal polynomial
14. Jordan canonical form

a. Python: NumPy and numerical linear algebra

Good basis
501 --> {502, 503, 504, 505}

Characteristic polynomial
506 --> 507

Diagonalization
508 --> 509 --> 510

Topics
511, 512, 513, 514, 50a

6. Theory of symmetric matrices

1. Reduction
2. Schur triangulation
3. Symmetric matrices and normal matrices
4. Spectral decomposition
5. Singular value decomposition
6. Understanding the principal component analysis
7. Inertia
8. Positivity
9. Rayleigh quotient
10. Cauchy interlacing theorem
11. Equitable partition
12. Covariance matrix
13. Principal component analysis
14. Laplacian matrix
15. Spectral embedding and clustering

Schur triangulation
601 --> 602 --> 603 --> 604 --> 605 --> 606

Symmetric matrix
607 --> 608 --> 609 --> 610 --> 611

Statistics
612 --> 613

Laplacian matrix
614 --> 615