James Cook's Linear Algebra Homepage:
Welcome. This webpage contains many resources I have created for linear algebra.

• Course Planner for Math 321 of Fall 2024


• You Tube Playlist for Fall 2024 Linear Algebra


• Lecture Notes for Math 321 of Fall 2024 ( posted 9-2-24)


• Assignments from Math 321 of Fall 2024:
• Mission 1
• Mission 2
• Quiz 2
• Mission 3
• Mission 4
• Quiz 4
• Bonus Project

General Advice: When confronting many "proof" problems in this course (and in more advanced abstract math courses) as a starting point you ought to ask yourself:

1. What am I asked to prove ?
2. Can I define the words used ?

It is not usually the case that you will find the same proof in my notes or some text. Definitions are key, I cannot emphasize this enough. Past this, you should consider using theorems, propositions etc. which we have developed. However, beware of proof by synonym. A common beginners mistake is to simply restate the claim in slightly different words as to prove the claim by invoking an entirely equivalent theorem. I usually write an arrow pointing back to itself to warn you of the circularity of such logic. Anyway, enough about what not to do, you can learn what is a good method of proof simply by following lectures and paying mind to study the structure of our arguments in each lecture. We do solve problems in this course, but, our more over-arching goal is to understand the structure of linear algebra.

Lectures from previous semesters:

• You Tube Playlist for Spring 2019 Linear Algebra
• You Tube Playlist for Spring 2018 Linear Algebra
• You Tube Playlist for Spring 2017 Linear Algebra
• You Tube Playlist from Linear Algebra of Spring 2016 (based on Curtis' text in part)

Useful Materials and Links:
This course is similar in spirit to previous offerings, however, I am making several significant modifications in format.

1. The websites linked below provide various matrix calculations online:
• Calculates the ref, rref, inverse and much more all while showing all the steps. very nice for problems without ugly decimals.
• Eigenvector calculator, ugly numbers no problem. Also deals with complex case no problem. However, does not find generalized e-vectors.
• Gram-Schmidt orthogonalizer: by Lawrence E. Turner of Southwestern Adventist University.
• Matrix Calculator Site - Calculates matrix inverse for you.
• Matrix Calculator Site - Multiplies matrices for you ( use to check answers )
• Wolfram Alpha: careful, may be addictive.
2. Previous versions of my notes are found below. Note, these are from the time when Math 221 was not a prerequisite for Math 321. The notes from 2024 and beyond have a significant rewrite.
   • Lecture Notes of 2019
   
   
   
   • Lecture Notes for Spring 2015
   • linear algebra (2010 version) paired with Lay's text.
   • linear algebra (2009 version) paired with Insel,Spence, Friedberg's basic text.
   • Lecture Notes for Applied Linear Algebra (2012 version)
3. The solutions below are from a previous semester.
   
   • Homework 1: on linear systems and Gaussian Elimination
   • Homework 2: matrix properties, elementary and inverse matrices
   • Homework 3: calculating inverse matrix, and determinants, Kramer's Rule
   • Homework 4: spanning sets, linear independence, CCP, coordinate vectors
   • Homework 5: row, column and null space of a matrix
   • Homework 6: linear transformations
   • Homework 7: dot-products, Gram Schmidt, orthogonal complements
   • Homework 8: orthogonal operators and least square fitting
   • Homework 9: inner products, real e-values and e-vectors
   • Homework 10: e-value theory and complex e-values and vectors
   • Homework 11: diagonalization and eigenbases
   • Homework 12: solving systems of ODEqns via the matrix exponential/e-vector method
4. An interesting example: the mapping T(A)=BAB has rank 4 despite the fact B has rank 2. See this row reduction which is in response to Part 2 of Lecture 20 of Linear Algebra 2015.

Spring 2015 Course Materials:
The content of 2016 and 2015 is more or less identical, however, I follow a somewhat different path. In particular, I have structured the quizzes to cover material in series rather than in parallel with the major tests. In any event, there is still value in last year's Lectures, but, I do hope this year is an improvement.

Spring 2015 Lectures, Quiz and Test Solutions:

1. Quiz 1 and its solution
2. Test 1 and its solution
3. Quiz 2 and its solution
4. Test 2 and its solution
5. Quiz 3 and its solution

Spring 2015 Lectures:

1. Lecture 1 part 1: sets, index notation, rows and columns
2. Lecture 1 part 2: equality by components, rows or columns
3. Lecture 2 part 1: functions, Gaussian elimination
4. Lecture 2 part 2: row reduction for solving linear systems
5. Lecture 3: solution sets, some theoretical results about rref
6. Lecture 4: rref pattern, fit polynomials, matrix algebra basics
7. Lecture 5: prop of matrix algebra, all bases belong to us, inverse matrix defined
8. Lecture 6: elementary matrices, properties and calculation of inv. matrix
9. Lecture 7: block-multiplication, (anti)symmetric matrices, concatenation
10. Lecture 8: span and column calculations in Rn, intro to LI
11. Lecture 9: LI and the CCP
12. Lecture 9 bonus: basics of linear transformations on Rn
13. Lecture 10: fundamental theorem of linear algebra (no video)
14. Lecture 11: gallery of LT, injectivity and surjectivity for LT, new LT from old
15. Lecture 12: examples and applications of matrices and LTs
16. Lecture 13 part 1: solution to Quiz 1
17. Lecture 13 part 2: solution to Quiz 1
18. Lecture 14: vector space defined, examples, subspace theorem
19. Lecture 15: axiomatic proofs, subspace thm proof, Null(A) and Col(A)
20. Lecture 16: generating sets for spans, LI, basis and coordinates
21. Lecture 17: theory of dimension and theorems on LI and spanning
22. Lecture 18: basis of column and null space, solution set structure again
23. Lecture 19: subspace thms for LT and unique linear extension prop
24. Lecture 19.5: isomorphism is equivalence relation, finite dimension classifies
25. Lecture 20 part 1: coordinate maps and matrix of LT for abstract vspace
26. Lecture 20 part 2: examples of matrix of LT in abstract case (Incidentally, my intuition at the end of this about the rank of the BAB mapping is incorrect. That map does in fact have rank 4 despite being built with the rank 2 B.)
27. Lecture 21: kernel vs nullspace, coordinate change
28. Lecture 22 part 1: coordinate change for matrix of LT
29. Lecture 22 part 2:rank nullity, Identity padded zeros thm, matrix congruence comment
30. Lecture 22.5: proof of abstract rank nullity theorem, examples
31. Lecture 23: part 1: quotient of vector space by subspace
32. Lecture 23 part 2: quotient space examples,1st isomorphism theorem
33. Lecture 24: structure of subspaces, TFAE thm for direct sums
34. Lecture 25: direct sums again, gallery of 3D isomorphic vspaces
35. Review for Test 2 part 1
36. Review for Test 2 part 2
37. Lecture 26: motivation, calculation and interpretation of determinants
38. Lecture 27: determinant properties, Cramer's Rule derived
39. Lecture 28: adjoint fla for inverse, eigenvectors and values
40. Lecture 28 additional eigenvector examples
41. interesting example for Lecture 28
42. Lecture 29: basic structural theorems about eigenvectors
43. supplement to Lecture 29
44. Lecture 30: eigenspace decompositions, orthonormality
45. concerning rotations
46. Lecture 31: complex vector spaces and complexification
47. Lecture 32: rotation dilation from complex evalue, GS example
48. Lecture 33: orthonormal bases, projections, closest vector problem.
49. Lecture 34: complex inner product space, Hermitian conjugate and properties
50. Lecture 35: overview of real Jordan form, application to DEqns
51. Lecture 36: invariant subspaces, triangular forms, nilpotentence
52. Lecture 37: nilpotent proofs, diagrammatics for generalize evectors, A = D + N
53. Lecture 38: minimal polynomial, help with homework
54. Lecture 39: solution to takehome Quiz 3
55. Lecture 40: partial course overview

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Last Modified: 9-27-2024