math 2301: intermediate linear algebra course themestpietrah/math2301/course_themes.pdf · linear...

Math 2301: Intermediate Linear AlgebraCourse Themes

Thom Pietraho

Bowdoin College

1 / 15Math 2301: Intermediate Linear Algebra , Course Themes

Linear Algebra

Initially: Linear algebra introduced as a convenient way to solve systems of equations.

Three sheaves of good crop, 2 sheaves of mediocre crop, and 1 sheaf of badcrop are sold for 39 dou. Two sheaves of good, 3 of mediocre, and 1 of badare sold for 34 dou. One sheaf of good, 2 of mediocre, and 3 of bad are soldfor 26 dou. What is the price for a sheaf of good crop, mediocre crop, andbad crop?

-Chiu-chang Suan-shu, Nine Chapters in Arithmetic (200 BC)

Today: Linear algebra is ubiquitous in mathematics and the sciences.

Course Goal: Introduce the mathematical tools and techniques necessary to be ableto use linear algebra in these contexts.

Linear Algebra

Matrices and Linear Transformations

Theme 1: Geometric meaning of matrices

This is a ”picture” of a 2x2 matrix:

−→

From this picture, one can glean its: eigenvalues , eigenvectors, rank, nullity,determinant .

Homework: Refresh your knowledge of these (briefly).

−→

From this picture, one can glean its: eigenvalues

, eigenvectors, rank, nullity,determinant .

−→

From this picture, one can glean its: eigenvalues , eigenvectors

, rank, nullity,determinant .

−→

From this picture, one can glean its: eigenvalues , eigenvectors, rank

, nullity,determinant .

−→

From this picture, one can glean its: eigenvalues , eigenvectors, rank, nullity

,determinant .

−→

From this picture, one can glean its: eigenvalues , eigenvectors, rank, nullity,determinant

−→

Bonus: Animation

Matrix Decompositions

Claim: The matrix:

−1 −1 10 1 11 0 1

is really the matrix

−1 0 00 −1 00 0 2

... in disguise.

Theorem

Every matrix is diagonal!

More precisely speaking, for any matrix M, we can always write

M = U · D · V

where D captures all the essential properties of M. The matrices U and V in somesense are “space-distorting glasses” that help you see this.

Claim: The matrix:

−1 −1 10 1 11 0 1

−1 0 00 −1 00 0 2

... in disguise.

Theorem

M = U · D · V

Claim: The matrix:

−1 −1 10 1 11 0 1

−1 0 00 −1 00 0 2

... in disguise.

Theorem

M = U · D · V

Claim: The matrix:

−1 −1 10 1 11 0 1

−1 0 00 −1 00 0 2

... in disguise.

Theorem

M = U · D · V

Application: Data Visualization

“Genes mirror geography in Europe,” Novembre et al, 2008.

Application: Clustering

Tissue samples, Uva et al. (dark blue = benign, red = normal.)

Application: Clustering

Voting records in Quebec.

Application: Pattern Fitting

Fitting a particular shape to a set of sample points.

Example: Linear regression in statistics. Many more.

Application: Pattern Fitting

Fitting a particular shape to a set of sample points.Example: Linear regression in statistics. Many more.

Norms and Inner Products

Question: Which matrix is a better approximation of(

1 23 4

(1.01 2.013.01 4.01

(1.04 2

To answer this question, we need a way of measuring distance between matrices.More importantly, this method should be useful in applications!

Caveat: Different applications have different demands. Need different ways ofmeasuring distance. This requires systematic study.

Foreshadowing: Analysis addresses this question in even more general contexts (notjust for matrices). But it is more “pretty” in linear algebra.

1 23 4

(1.01 2.013.01 4.01

(1.04 2

1 23 4

(1.01 2.013.01 4.01

(1.04 2

1 23 4

(1.01 2.013.01 4.01

(1.04 2

Application: Image Compression

This is a matrix:

This one is an approximation:

Question: How to find the best approximation using the least amount of data?

This is a matrix:

Application: Sound Compression

This sounds very similar to the question above, but the theory is much different. In fact,requires a vector space very different from Rn.

Moral: As the class goes on, some of our results will be done in more general settingthan Rn.

This sounds very similar to the question above, but the theory is much different.

In fact,requires a vector space very different from Rn.

Other Vector Spaces

Rn: vectors are n-tuples of numbersApplications: many

C[a, b] and P[a, b] : vectors are functions or polynomialsApplications: statistics, partial differential equations, data filters.

S({0, 1}): vectors are sequences of 0s and 1s.Applications: digital data, wavelets

Upshot: A theorem useful to us in Rn many have interesting implications in these othercontexts. What is a matrix? What does it mean for it to be “diagonal”?

Other Vector Spaces

Networks

Eileen Brown’s Social Network:

This can be encoded by a matrix.

Upshot: Can use linear algebra to analyze the graph.Questions: Find cliques, model disease propagation.Question: How to graphically render the graph to display most information?

Networks

This can be encoded by a matrix.Upshot: Can use linear algebra to analyze the graph.

Questions: Find cliques, model disease propagation.Question: How to graphically render the graph to display most information?

Networks

This can be encoded by a matrix.Upshot: Can use linear algebra to analyze the graph.Questions: Find cliques, model disease propagation.

Question: How to graphically render the graph to display most information?

Networks

This can be encoded by a matrix.Upshot: Can use linear algebra to analyze the graph.Questions: Find cliques, model disease propagation.Question: How to graphically render the graph to display most information?

Neural Networks and Machine Learning

An Artificial Neural Network:

Machine learning employs a considerable amount of linear algebra, and it turns out thatsometimes linear algebra can be learned by neural networks. But more on that later . . .Image: medium.com

math 2301: intermediate linear algebra course themestpietrah/math2301/course_themes.pdf · linear...

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