MATH 341 lecture 005 fall 2020

Homework solutions & comments

• Homework 6
• About the 3rd midterm
• About the 2nd midterm
• Homework 5
• Homework 4
• Homework 3
• About the 1st midterm
• Homework 2
• Homework 1

Course notes

We will follow the book Linear Algebra by Friedberg, Insel and Spence (4^th edition). In addition here are my shorter notes on the chapters in the book:

• About sets and the axiomatic method
• Vector Spaces
• Linear Transformations
• Determinants
• Eigenvectors and values
• Inner product spaces

Lecture notes

An after the fact short summary of each lecture. For pdf files containing the written part of each lecture look under ‘Files’ on canvas.

December

Thu 10 Symmetric linear transformations

Tue 8 Inner product spaces

Thu 3 Diagonalization

Tue 1 Eigenvalues and eigenvectors

November

Tue 24 Midterm 3

Thu 19 Cramer's rule. Determinant of the product of two matrices.

Tue 17 Determinants and row/column expansion. The cofactor matrix. Cramer's rule

Thu 12 Q&A about hwk 5.

Tue 10 Determinants.

Thu 5 The space of linear transformations $\mathcal L(V)$. Inverses and invertibility.

Tue 3 The matrix of a linear transformation. Examples: rotations and reflections in the plane with respect to different bases. $AB$ and $BA$ don't have to be the same.

October

Thu 29 Midterm 2.

Tue 27 Review of problems 2, 5 from hwk 4. Linear transformations of $\R^n$ to $\R^m$. Matrices and matrix multiplication

Thu 22 Some more facts about bases and dimensions of subspaces. Solving systems of linear equations; computational examples

Tue 20 Linear transformations: null space and range. Statement and proof of Rank+Nullity theorem. Injectivity Theorem, Bijectivity Theorem.

Thu 15 Linear transformations: definition, examples from geometry, algebra, differential operators

Tue 13 Proof by mathematical induction and a proof of the dimension theorem.

Thu 8 Bases: definition and examples in $\mathbb{F}^n$, the space of polynomials, and $\R$ as a vector space over the field $\mathbb{Q}$.

Tue 6 Midterm 1

Thu 1 Comments on the midterm, see also here. More examples of linear independence for vectors in $\R^2, \R^4$, $\mathcal{P}_3(\R)$, and $\mathcal{F}(\{0,1,2\},\R)$.

September

Tue 29: Theorem on the intersection of subspaces being itself a subspace. Linear independence: definition and examples. Definition of a basis.

Thu 24: Examples of linear combinations, span of vectors, examples in the space of polynomials. Systematically solving systems of linear equations via row reduction: examples.

Tue 22: The space of functions on an interval and the set of solutions to a linear differential equation. Linear combinations, span of a set.

Thu 17: Examples of subspaces, including lines in the plane. Proof by contradiction. Examples of using the axioms and theorems to prove that a given subset $W\subset V$ of a vector space $V$ is a linear subspace. The space of symmetric matrices. The space of polynomials of degree at most $N$.

Tue 15: The space of polynomials; the space of trigonometric polynomials; linear subspaces.

Thu 10: Class was cancelled due to covid.

Tue 8: We discussed some first proofs of facts that follow from the Vector Space Axioms. At the end of the class I began describing new examples of vector spaces by introducing $\mathcal M_{m\times n}(\mathbb{F})$, the space of $m\times n$ matrices with coefficients from a field $\mathbb{F}$. Next time we will see more examples, notably the space of polynomials, and the space of trigonometric polynomials, and function spaces.
We will then continue with linear subspaces.

Thu 3: In the first lecture I introduced the axioms of a vector space, and emphasized that you have to say which field of numbers can multiply the vectors. Then I described the examples $\R^n$, $\mathbb{C}^2$, and ended by noting that we can consider $\R$ as a vector space where the number field $\mathbb{F}$ is $\mathbb{Q}$.

The goal for Tuesday is to do some first proofs and discuss more examples of vector spaces.