Linear Algebra I --- Collaborative notes, 2022FMath103A

--- description: notes, 2022F, math103a, linear algebra, jephian, nsysu, 線性代數, 林晉宏 tags: learning-together --- {%hackmd 5xqeIJ7VRCGBfLtfMi0_IQ %} :::success **Tasks** :loudspeaker: :loudspeaker: :loudspeaker: **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(c), 7(a), 7(b) - 102: 4 - 103: 4(b), 4(c), 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(c), 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(c) - 203: 4, 5(a), 6, 7 - 204: 6 - 205: 4, 5 - 206: 3, 4, 5, 6 - 207: 4(b), 4(c), 4(d), 5(a), 5(b) - 208: 1(c), 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](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: - 101: 2(鄭宗祐 +2), 5(b)(鄭宗祐), 6(a)(鄭宗祐), 6(c)(鄭宗祐), 7(c)(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(c)(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. ::: :::success Math Runway Exercise by November 30. You will get 1 extra point if you finished one exercise correctly. :100: 鄭宗祐, 湯昆璋, 黃經幃 ::: # Linear Algebra I --- Collaborative notes, 2022FMath103A  ### 1. Linear geometry 1. [Vector, length, and angle](https://hackmd.io/BbVvTBUGTTmGV-5p8IQbUA) 2. [Subspaces in Rn](https://hackmd.io/cjJtz6dSStqNFasQUz8otg) 3. [Column space of a matrix](https://hackmd.io/4vkTUpZ2ST2HWMhu4SBKGA) 4. [Row space of a matrix](https://hackmd.io/mh0rmua1R4upAdCZ2cOizw) 5. [Projection and reflection](https://hackmd.io/ApSNHdZBTxanm8fYq3oiJQ) 6. [Affine subspaces in Rn](https://hackmd.io/8uqUWKvSTBiA0KaEZp1ifg) 7. [Solution set of Ax = b](https://hackmd.io/FW6pNT4nQ3CgK_zXRkb-HQ) 8. [Row operations](https://hackmd.io/FDYjsnzGQ1uWjYQWlGJ8ug) 9. [Finding a particular solution](https://hackmd.io/Ko171PehQyyrrFg0V6jRnA) 10. [Finding the homogeneous solutions](https://hackmd.io/NkSxKEh4Q_m7sQgmBdMxJA) 11. [Number of solutions](https://hackmd.io/oC1HF81vR825xlTuI3UqtA) 12. [Matrix inverse](https://hackmd.io/Pv9Er-LXT3Cvy99-qJrrPA) 13. [Elementary matrices](https://hackmd.io/PIRAcXdcQYOMBj7CGM7RDw) 14. [Four fundamental subspaces](https://hackmd.io/mREwSBslTACYt-jRhycF5A) a. [Sage: Matrices and linear equations](https://hackmd.io/KFJoxQqATEqbPhwvmEAp4A) ``` 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](https://hackmd.io/jdcuVbBJTTqXle1JWAmFzw) 2. [Basis](https://hackmd.io/HqDC9Eq6QVS-0wgZsBigCQ) 3. [Column space, left kernel, and their bases](https://hackmd.io/yYopnAOfTYqxcTiBLBIhzQ) 4. [Row space, kernel, and their bases](https://hackmd.io/kBd4_wDWT9a-0Tn6lTumzQ) 5. [Basis exchange lemma](https://hackmd.io/V8SEn0dHQ0CGQ3OBkxEWpQ) 6. [Dimension, expanding and shrinking lemmas](https://hackmd.io/JnFHKaufRs-9IdWzb_RMUg) 7. [Rank and nullity](https://hackmd.io/iHYhjee0SKCNOaBGJsWpsw) 8. [Vector space](https://hackmd.io/0j2iAezNQk29zJOJcHIX3Q) 9. [Subspaces in a vector space](https://hackmd.io/KHGk9oCEQqyQlLanNS-E0g) 10. [Common vector spaces](https://hackmd.io/PqHPdfCFRi2zd1IeYq66SA) 11. [Constructing new subspaces](https://hackmd.io/jhk026nlR9-fgussVAE5-Q) 12. [Constructing new vector spaces](https://hackmd.io/KetKM9EFQZ63aHoWPlLgaQ) 13. [Orthogonal geometry](https://hackmd.io/K1JKcxawQImBO2OOEh_O4w) 14. [Gram–Schmidt orthogonalization](https://hackmd.io/T0MB5E0OQ5SMwH3s_oTziQ) 15. [Direct sum of orthogonal subspaces](https://hackmd.io/xywvewKbTc-0xIv2GDgwYw) ``` 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](https://hackmd.io/qvQwmOUkRYCjlxQOsoLNZQ) 2. [Linear function](https://hackmd.io/5KlvuwTtQ4m97JyJCgtA7A) 3. [Matrix as a linear function](https://hackmd.io/ETQMq1tjTn6qtG1XmXpaDg) 4. [Linear function as a matrix](https://hackmd.io/iPHUR2USRzaspdwC408bYA) 5. [Vector representation in Rn](https://hackmd.io/3rSy-rwDRteRjowQNgbS_A) 6. [Vector representation in a vector space](https://hackmd.io/6xe-kjIMRN6npNcRw56tGg) 7. [Change of basis](https://hackmd.io/S8Bh-QN2QwuPtqIQwJKo1A) 8. [Isomorphism](https://hackmd.io/0376Q7JlQmeYMTULcgYzaw) 9. [Matrix representation in Rn](https://hackmd.io/6c8i06EfQvmwpauOGp3QSw) 10. [Matrix representation in a vector space](https://hackmd.io/J6yRbJbRSPu_QG-M4W88uA) 11. [Lagrange polynomials and Vandermonde matrix](https://hackmd.io/tUuam7BAS0mcJdHw4BWAFw) 12. [Sylvester matrix and resultant](https://hackmd.io/nipjlKuXRFylzFKgfLYkNw) 13. [Understanding the spectral decomposition](https://hackmd.io/uGgY48wESaO6CTj95t0KTg) 14. [Understanding the singular value decomposition](https://hackmd.io/diWni-zVSxiSXltIylVyFg) 15. [Understanding the Jordan canonical form](https://hackmd.io/n86YO2miRGmKGVDVbsualA) ``` Linear function 301 --> 302 --> 303 -->304 Vector and matrix representations 305 --> 306 --> 307 --> 308 --> 309 --> 310 Topics 311, 312, 313, 314, 315 ```