Linear Algebra II --- Collaborative notes, 2023SMath104A

--- description: notes, 2023S, math104a, linear algebra, jephian, nsysu, 線性代數, 林晉宏, 中山 tags: learning-together --- {%hackmd 5xqeIJ7VRCGBfLtfMi0_IQ %} :::success **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. :warning: 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](https://hackmd.io/) 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. ```markdown {%hackmd 5xqeIJ7VRCGBfLtfMi0_IQ %} ``` Thus, you may use some macros, such as $\bv$. 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.) ::: :::success Finished Tasks: :thumbsup: :thumbsup: :thumbsup: - 401: 3 (:cloud:) - 402: 1(b) (鄭宗祐), 2a (:cloud:), 3b (:cloud:), 4 (:cloud:), 5 (:cloud:) - 403: 6 (:cloud:) - 404: 3 (鄭宗祐) - 405: 3(b) (:cloud:), 5(a,b) (:cloud: +1), 6(a,b,c) (:cloud: +1.5) - 406: 4(a,b,c) (:cloud: +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) (:cloud: +1.5), 2 (:cloud: +1), 3(a,b,c,d) (:cloud: +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](https://hackmd.io/@jephianlin/2022FMath103A-notes) ### 4. Determinant 1. [Determinant for small matrices](https://hackmd.io/6Y3sPfVbTnuvrvtvhtwgCA) 2. [Invertible matrix as the product of elementary matrices](https://hackmd.io/pTRdF8YhT2G3KAkSjJePUg) 3. [Elementary matrix acts on columns](https://hackmd.io/YkT4UjgoSKa4nz2GwTnkKw) 4. [Elementary matrix acts on rows](https://hackmd.io/BLY5xaImSninJtb48Hytnw) 5. [Definition of the determinant](https://hackmd.io/08YFs4gtTxud7fJw8G1mkg) 6. [Invertibility](https://hackmd.io/EaKJRgsmRn-njeJJdEow-g) 7. [Matrix multiplication and transpose](https://hackmd.io/8nlr7aJ-R7y7GhTOr3wYeg) 8. [Block matrix](https://hackmd.io/l9yF9JW0SgmWvWQLsuDwPw) 9. [Distributive law and Laplace expansion](https://hackmd.io/nIiK8CjpQO-E0adtTvJqUg) 10. [Adjugate](https://hackmd.io/b0zRpo0KS0S6MlrID9v2aw) 11. [Cramer's rule](https://hackmd.io/b2MCRSmyS4qbdlGy8cWAsQ) 12. [Permutation matrix](https://hackmd.io/SKNNl8AIR8eMkFIo8fqcEQ) 13. [Permutation expansion](https://hackmd.io/B_k6-mZdQv6pOgiVFHRVjA) 14. [Interpretation through graph theory](https://hackmd.io/g2Oqdrw8TlmhI1jNaoeBPQ) 15. [Determinant is well-defined](https://hackmd.io/9wUjdTwoS_KlaoDualS1jg) ``` 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](https://hackmd.io/EBAMw13tTLaUJCV62-hCiw) 2. [Quadratic curve](https://hackmd.io/LaagNizzSCaEmKDUl2qN5g) 3. [Recurrence relation](https://hackmd.io/NO4tWmFPS3mNj7PwPYTzxg) 4. [Linear differential equation](https://hackmd.io/EPjRx76SSKm-TZ6uBUmzlg) 5. [Matrix exponential](https://hackmd.io/LfCR00utQU-hxgsMBYXdxw) 6. [Characteristic polynomial](https://hackmd.io/4MstlCgGTFC-bg1rzRz9Xg) 7. [Coefficients of the characteristic polynomial](https://hackmd.io/gDog2I0iTce8a1ow2YaYuQ) 8. [Diagonalization](https://hackmd.io/lbjUOXQRRR6ar9dB5llxGw) 9. [Algebraic multiplicity and geometric multiplicity](https://hackmd.io/RBqofwTuRXyVUbNWw7nDUw) 10. [Eigenspace](https://hackmd.io/_ZHZ027GQDmDPVcsNlYcMw) 11. [Graph and characteristic polynomial](https://hackmd.io/YtinC7pLQeWgaoPwaN7Xuw) 12. [Cayley–Hamilton theorem](https://hackmd.io/2jwHt8XjQtaSFUIgzhEo7g) 13. [Minimal polynomial](https://hackmd.io/n0w5ic4cTOmU6-QSRbTDtQ) 14. [Jordan canonical form](https://hackmd.io/8Q3DwhFkR_aRPSP2aIvd7A) a. [Python: NumPy and numerical linear algebra](https://hackmd.io/VEtdgc98RG62wlm2Ek7adA) ``` 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](https://hackmd.io/nbt7o_mvQc2NLu-taxbsxw) 2. [Schur triangulation](https://hackmd.io/LjrcgX0_TJ6YzJpvpPD8AQ) 3. [Symmetric matrices and normal matrices](https://hackmd.io/mDM4c-nFSw-dNJ5UjIey3A) 4. [Spectral decomposition](https://hackmd.io/fCAfBvPITAWO-q_NYqOf9g) 5. [Singular value decomposition](https://hackmd.io/j1H2Sl_TQWC3yjcse2ah6g) 6. [Understanding the principal component analysis](https://hackmd.io/S-i_dV07SUKmL88rSPnSNg) 7. [Inertia](https://hackmd.io/r7pJKs6CSICHMsp-aEYQCw) 8. [Positivity](https://hackmd.io/myWRFbUpQ9-yvV_FH65OWA) 9. [Rayleigh quotient](https://hackmd.io/5YYCjhXaT6yi3o06vb39wA) 10. [Cauchy interlacing theorem](https://hackmd.io/Wv5BDL8NR_GeHvZBD6KouQ) 11. [Equitable partition](https://hackmd.io/zKgQNjrCR4-UY7mbWB-BOQ) 12. [Covariance matrix](https://hackmd.io/yGHNUdQeTqiFdysKixDuNw) 13. [Principal component analysis](https://hackmd.io/iWjQw4R9QOyMJYq-22P8RA) 14. [Laplacian matrix](https://hackmd.io/Rv-XadyPQ7a2agSW4tAuhg) 15. [Spectral embedding and clustering](https://hackmd.io/-UD7c-RPRsywbkJK2zhLGQ) ``` Schur triangulation 601 --> 602 --> 603 --> 604 --> 605 --> 606 Symmetric matrix 607 --> 608 --> 609 --> 610 --> 611 Statistics 612 --> 613 Laplacian matrix 614 --> 615 ```