James Cook's Introduction to Linear Algebra Page
James Cook's Linear Algebra Homepage:
Welcome. This webpage contains many resources I have created for linear algebra. The current semester work is found in Blackboard, thanks!
When confronting many "proof" problems in this course (and in more advanced abstract math courses) as a starting point you ought to ask yourself:
- 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)
- Homework is due about every week. The wise course of action is to work ahead and ask questions in office hours. You can work together, but
in the end your solution must be your own.
- I'm here to help. Please make wise use of my office hours and the help session when you get stuck. Also, in lecture, if I write something obviously wrong. Please politely interupt me before I burn 5 minutes of class on a bogus calculation. Do not wait until the end of class to tell me as we exit, you wrote linearly independent but meant linearly dependent. I have no patience for corrections! I want them immediately.
- Someone will be unable to make it to my office hours. It's inevitable in a given semester. Therefore, if you are such a person, it especially important for you to work with a study group which has at least one person who can make it to office hours.
- Homework will initially be posted in Blackboard. I don't remind you when it is due since the Course Planner (see above) has the due dates. Late work is not generally accepted.
- relax, it's just algebra!
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.
- What am I asked to prove ?
- Can I define the words used ?
Spring 2019 Lecture Notes:
Useful Materials and Links:
This course is similar in spirit to previous offerings, however, I am making several significant modifications in format.
Previous versions of my notes are found below. The fourth version of the 321 notes is what I expect you to read this semester along side the required text. I will post the notes as I write them throughout the semester (in Blackboard, I will announce when an update is made)
The solutions below are from a previous semester.
- The websites linked below provide various matrix calculations online:
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:
- 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
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:
Spring 2015 Lectures:
- Quiz 1
and its solution
- Test 1
and its solution
- Quiz 2
and its solution
- Test 2
and its solution
- Quiz 3
and its solution
Lecture 1 part 1: sets, index notation, rows and columns
- Lecture 1 part 2: equality by components, rows or columns
- Lecture 2 part 1: functions, Gaussian elimination
- Lecture 2 part 2: row reduction for solving linear systems
- Lecture 3: solution sets, some theoretical results about rref
- Lecture 4: rref pattern, fit polynomials, matrix algebra basics
- Lecture 5: prop of matrix algebra, all bases belong to us, inverse matrix defined
- Lecture 6: elementary matrices, properties and calculation of inv. matrix
- Lecture 7: block-multiplication, (anti)symmetric matrices, concatenation
- Lecture 8: span and column calculations in Rn, intro to LI
- Lecture 9: LI and the CCP
- Lecture 9 bonus: basics of linear transformations on Rn
- Lecture 10: fundamental theorem of linear algebra (no video)
- Lecture 11: gallery of LT, injectivity and surjectivity for LT, new LT from old
- Lecture 12: examples and applications of matrices and LTs
- Lecture 13 part 1: solution to Quiz 1
- Lecture 13 part 2: solution to Quiz 1
- Lecture 14: vector space defined, examples, subspace theorem
- Lecture 15: axiomatic proofs, subspace thm proof, Null(A) and Col(A)
- Lecture 16: generating sets for spans, LI, basis and coordinates
- Lecture 17: theory of dimension and theorems on LI and spanning
- Lecture 18: basis of column and null space, solution set structure again
- Lecture 19: subspace thms for LT and unique linear extension prop
- Lecture 19.5: isomorphism is equivalence relation, finite dimension classifies
- Lecture 20 part 1: coordinate maps and matrix of LT for abstract vspace
- 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.)
- Lecture 21: kernel vs nullspace, coordinate change
- Lecture 22 part 1: coordinate change for matrix of LT
- Lecture 22 part 2:rank nullity, Identity padded zeros thm, matrix congruence comment
- Lecture 22.5: proof of abstract rank nullity theorem, examples
- Lecture 23: part 1: quotient of vector space by subspace
- Lecture 23 part 2: quotient space examples,1st isomorphism theorem
- Lecture 24: structure of subspaces, TFAE thm for direct sums
- Lecture 25: direct sums again, gallery of 3D isomorphic vspaces
- Review for Test 2 part 1
- Review for Test 2 part 2
- Lecture 26: motivation, calculation and interpretation of determinants
- Lecture 27: determinant properties, Cramer's Rule derived
- Lecture 28: adjoint fla for inverse, eigenvectors and values
- Lecture 28 additional eigenvector examples
- interesting example for Lecture 28
- Lecture 29: basic structural theorems about eigenvectors
- supplement to Lecture 29
- Lecture 30: eigenspace decompositions, orthonormality
- concerning rotations
- Lecture 31: complex vector spaces and complexification
- Lecture 32: rotation dilation from complex evalue, GS example
- Lecture 33: orthonormal bases, projections, closest vector problem.
- Lecture 34: complex inner product space, Hermitian conjugate and properties
- Lecture 35: overview of real Jordan form, application to DEqns
- Lecture 36: invariant subspaces, triangular forms, nilpotentence
- Lecture 37: nilpotent proofs, diagrammatics for generalize evectors, A = D + N
- Lecture 38: minimal polynomial, help with homework
- Lecture 39: solution to takehome Quiz 3
- Lecture 40: partial course overview
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Last Modified: 1-3-2019