Lectures in TMA4115 - Calculus 3, spring 2013 - MTFYMA

Below is a description of what I intend to cover in each lecture and slides from the lectures.

I will try to make slides available before the lectures so you can print them and follow along during the lectures. Slides with extra details (for instance examples and proofs discussed in the lectures) will be made available after the lectures.

If I find mistakes on the slides during the lectures, I will correct these mistakes before I upload the slides with extra details, but I will probably not changes the slide I already have made available on the web page.

The slides with extra details will come in two versions. One with transitions (these might be useful for viewing on a screen), and one without transitions (best for printing).

Lecture 1, January 16

I will give a short introduction to the course and to the course web page. I will then introduce complex numbers and tell how they can be represented as points in the plane, how we add and multiply complex numbers and tell what complex conjugation is.

Pages ix-xiv in the book.

Lecture 2, January 17

I will give a review of what we looked at in lecture 1 and then tell about polar representation of complex numbers, de Moivre's Theorem, root of complex numbers, and roots of unity.

Pages ix-xviii in the book.

Lecture 3, January 23

In this lecture we shall

  • use complex numbers to solve polynomial equations,
  • look at the fundamental theorem of algebra,
  • introduce the complex exponential function,
  • and study extensions of trigonometric functions to the complex numbers.

Sections 4.1 and 4.3 in “Second-Order Equations” (pages xxxv-xlv and xlix-lv).

Lecture 4, January 24

In this lecture we shall

  • study second-order linear differential equations,
  • introduce the Wronskian,
  • completely solve second-order homogeneous linear differential equations with constant coefficients.

Sections 4.1 and 4.3 in “Second-Order Equations” (pages xxxv-xlv and xlix-lv).

Lecture 5, January 30

In this lecture we shall

  • study harmonic motions,
  • study solutions of second-order linear inhomogeneous differential equations,
  • look at the method of undetermined coefficients.

Sections 4.4 and 4.5 in “Second-Order Equations” (pages pages lv–lxxii).

Lecture 6, January 31

In this lecture we shall

  • look at variation of parameters,
  • study forced harmonic motions.

Sections 4.6 and 4.7 in “Second-Order Equations” (pages pages lxxii–lxxxvi).

Lecture 7, February 6

In this lecture we shall

  • study how to solve systems of linear equations,
  • introduce row reduction, echelon forms, pivot positions, the row reduction algorithm, and parametric descriptions of solution sets of systems of linear equations.

Sections 1.1-1.2 in “Linear Algebras and Its Applications” (pages 1-23).

Lecture 8, February 7

In this lecture we shall introduce and study

  • vectors,
  • linear combinations of vectors,
  • subsets spanned by vectors,
  • vector equations,
  • the product of a matrix and a vector,
  • matrix equations.

Sections 1.3-1.4 in “Linear Algebras and Its Applications” (pages 24-42).

Lecture 9, February 13

In this lecture we shall

  • introduce and solve homogeneous and nonhomegeneous matrix equations,
  • learn how to write solution sets in parametric vector form,
  • look at applications of linear systems.

Sections 1.5-1.6 in “Linear Algebras and Its Applications” (pages 43-55).

Lecture 10, February 14

In this lecture we shall

  • define and study linear independence of vectors.

Section 1.7 in “Linear Algebras and Its Applications” (pages 55-62).

Lecture 11, February 20

In this lecture we shall introduce and study

  • linear transformations,
  • the standard matrix of a linear transformation,
  • onto linear transformations,
  • one-to-one linear transformations.

Sections 1.8-1.9 in “Linear Algebras and Its Applications” (pages 62-80).

Lecture 12, February 21

In this lecture we shall

  • look at applications of linear models,
  • look at the use of Maple and WolframAlpha.

Section 1.10 in “Linear Algebras and Its Applications” (pages 80-90).

Lecture 13, February 27

In this lecture we shall

  • see how we can add and multiply matrices,
  • define and study invertible matrices.

Sections 2.1–2.2 in “Linear Algebras and Its Applications” (pages 91–111).

Lecture 14, February 28

In this lecture we shall

  • further study invertible matrices,
  • look at the invertible matrix theorem.

Section 2.3 in “Linear Algebras and Its Applications” (pages 111–116).

Lecture 15, March 6

In this lecture we shall

  • introduce and study determinants.

Sections 3.1–3.2 in “Linear Algebras and Its Applications” (pages 163–177).

Lecture 16, March 7

In this lecture we shall

  • look at Cramer’s rule,
  • give a formula for the inverse of an invertible matrix,
  • look at the relationship between areas, volumes and determinants.

Section 3.3 in “Linear Algebras and Its Applications” (pages 177–187).

Lecture 17, March 13

In this lecture we shall introduce and study

  • abstract vector spaces and subspaces,
  • null spaces and column spaces of matrices,
  • linear transformations between abstract vector spaces,
  • kernels and ranges of linear transformations,
  • linear independence and linear dependence in abstract vector spaces.

Sections 4.1–4.3 in “Linear Algebras and Its Applications” (pages 189–209).

Lecture 18, March 14

In this lecture we shall introduce and study

  • bases of vector spaces,
  • coordinate systems in vector spaces relative to bases.

Section 4.3–4.4 in “Linear Algebras and Its Applications” (pages 209–225).

Lecture 19, March 20

In this lecture we shall introduce and study

  • the dimension of a vector space,
  • the rank of a matrix.

Sections 4.5–4.6 in “Linear Algebras and Its Applications” (pages 225–238).

Lecture 20, March 21

In this lecture we shall look at

  • applications to Markov chains.

Section 4.9 in “Linear Algebras and Its Applications” (pages 253–264).

Lecture 21, April 3

In this lecture we shall introduce and study

  • eigenvectors, eigenvalues and eigenspaces of square matrices,
  • the characteristic polynomial of a square matrix.

Sections 5.1-5.2 in “Linear Algebras and Its Applications” (pages 265–281).

Lecture 22, April 4

In this lecture we shall look at

  • similar matrices,
  • diagonal matrices,
  • diagonalizable matrices.

Section 5.3 in “Linear Algebras and Its Applications” (pages 281–288).

Lecture 23, April 10

In this lecture we shall study

  • real matrices with complex eigenvalues,
  • systems of first order differential equations.

Sections 5.5 and 5.7 in “Linear Algebras and Its Applications” (pages 295-301, and 311-319), and Section 4.2 in “Second-Order Equations” (pages xlv-xlix).

Lecture 24, April 11

In this lecture we shall continue studying systems of first order differential equations.

Section 5.7 in “Linear Algebras and Its Applications” (pages 295-301, and 311-319), and Section 4.2 in “Second-Order Equations” (pages xlv-xlix).

Lecture 25, April 17

In this lecture we shall introduce and study

  • the inner product,
  • the length of a vector,
  • orthogonality and orthogonal sets in \(\mathbb{R}^n\),
  • orthogonal and orthonormal bases,
  • the orthogonal complement of a subspace,
  • orthogonal matrices.

Sections 6.1–6.2 in “Linear Algebras and Its Applications” (pages 329-346).

Lecture 26, April 18

In this lecture we shall introduce and study

  • orthogonal projections,
  • the Gram-Schmidt process,
  • QR factorization.

Sections 6.3–6.4 in “Linear Algebras and Its Applications” (pages 347-360).

Lecture 27, April 24

In this lecture we shall look at

  • least-squares problems,
  • applications to linear models.

Sections 6.5–6.6 in “Linear Algebras and Its Applications” (pages 360–375).

Lecture 28, April 25

In this lecture we shall introduce and study

  • symmetric matrices,
  • quadratic forms.

Sections 7.1–7.2 in “Linear Algebras and Its Applications” (pages 393–407).

2013-04-25, tokemeie