Upper level courses I teach include Differential Geometry MAT 4140, Senior Capstone MAT 4040, Instructional Assistant MAT 3520

How are addition and scalar multiplication important in linear algebra?

What are some topics where we've seen algebra and geometry perspectives

Why do you think critical analysis and reasoning are a focus in this course?

What are applications we've seen?

Work on the research presentations.

We have been investigating applications of chapter 5 material to mathematical biology. We'll finish the semester by looking (briefly) at applications of eigenvalues and eigenvectors to computer science and physics:

Computer Graphics: Look at MatrixInverse(P).A.P, which has the eigenvalues on the diagonal - definition of similarity, like in 6.1 in the book.

Execute in Maple:

A:=Matrix([[(cos(theta))^2,cos(theta)*sin(theta)],[cos(theta)*sin(theta), ((sin(theta))^2)]]);

h,P:=Eigenvectors(A)

Diag:=simplify(MatrixInverse(P).A.P);

What geometric transformation is Diag?

Notice that P.Diag.MatrixInverse(P) = A by matrix algebra.

Writing out a transformation in terms of a P, the inverse of P, and a diagonal matrix is very useful in computer graphics [Recall that we read matrix composition from right to left].

Linear Transformations

Mathematical Physics:

heat diffusion (48 seconds)

Eigenfunctions and the heat equation

Mention the spectrum of the Laplacian [divergence of gradient] (the discrete sequence of eigenvalues). Spectrum is applied to graphs, the heat equation,...

Can you hear the shape of a drum?

Sound of quantum drums

END OF MATERIAL FOR TEST 3

Chap 5 clicker questions#4-8

Take any questions. Finish the dynamical systems demo. Compute the eigenvalues using determinant(A-lambdaI)=0

Clicker review of 2.8, 5.1 and 5.6:

#1, 5-8 in 2.8 clicker questions.

Chap 5 clicker questions#1 and #3

Geometry of Eigenvectors and compare with Maple.

Eigenvector decomposition for a diagonalizable matrix A_nxn [where the eigenvectors form a basis for all of R

Foxes and Rabbits demo on ASULearn

Dynamical Systems and Eigenvectors on ASULearn

If ___ equals 0 then we die off along the line____ [corresponding to the eigenvector____], and in all other cases we [choose one: die off or grow or hit and then stayed fixed] along the line____ [corresponding to the eigenvector____].

clicker questions on inverses and determinants #3-4 and 6-8

Review and finish 2.8 using the matrix 123,456,789 and finding the Nullspace and ColumnSpace (using 2 methods - reducing the spanning equation with a vector of b1...bn, and separately by examining the pivots of the ORIGINAL matrix.)

Define eigenvalues and eigenvectors [Ax=lambdax, vectors that are scaled on the same line through the origin, matrix multiplication is turned into scalar multiplication].

Eigenvectors of Matrix([[0,0],[1,0]]);

Algebra: Show that we can solve using det(A-lambdaI)=0 and (A-lambdaI)x=0.

Compute the eigenvectors of Matrix([[0,1],[1,0]] by-hand and compare with Maple's work.

Review the 2 determinant methods for the 123,456,789 matrix. Show that for 4x4 matrix in Maple, only Laplace's method will work.

The connection of row operations to determinants

clicker questions on inverses and determinants #3-5

Continue determinant work via the relationship of row operations to the geometry of determinants via a demo on ASULearn.

Show that det A non-zero can be added into Theorem 8 in Chapter 2.

Algebraic and geometric ideas related to the determinant, including the determinant of A inverse, A transpose and A triangular (such as in Gaussian form).

2.3 and linear transformation Clicker questions #8-9

End of Computer Graphics Demo - rotating a 3-d object.

Computer graphics continued, including the benefit of derivatives and unit length vectors in keeping a car on a race track - demo on ASULearn.

Discuss Yoda via the file yoda2.mw with data from Lucasfilm LTD as on Tim's Page which has the data.

Begin chapter 3 via mentioning google searches:

application of determinants in physics

application of determinants in economics

application of determinants in chemistry

application of determinants in computer science

Eight queens and determinants

Chapter 3 in Maple via MatrixInverse command for 2x2 and 3x3 matrices and then determinant work, including 2x2 and 3x3 diagonals methods, and Laplace's expansion (1772 - expanding on Vandermonde's method) method in general. [general history dates to Chinese and Leibniz]

2.3 and linear transformation Clicker questions #1-7

general geometric transformations on R

Computer graphics Demo on ASULearn [2.7]

A_mxn (not square). Can Ax=0 have only the trivial soluiton

a) No that statement is impossible

b) Yes when the columns of A are l.i.

c) Yes when the columns of A are l.i. and A has m pivot rows

d) Yes when the columns of A are l.i. and A has n pivot columns

e) Both c and d

Review guidelines for Problem Sets, including

Computer graphics and linear transformations (1.8, 1.9, 2.3 and 2.7)

Begin with dilations

Guess the transformation on ASULearn

Revisit Theorem 8 in 2.3 by incorporating the language of linear transformations [while also covering 1-1 and onto material in 1.9]

Review the unit circle

2.3 clicker questions

2.2 and 2.3 clicker questions

Theorem 8 in 2.3 [without linear transformations]

Catalog description:

-2.1-2.3 Applications: Coding, Condition Number and Linear Transformations (2.3, 1.8, 1.9 and 2.7)

-Chapter 3 determinants and applications

-Eigenvalues and applications (2.8, 4.9 and chap 5 selections, 7.1... as time allows)

-Final research sessions [research a topic related to the course that you are interested in]

Hill Cipher

A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X | Y | Z |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 |

Condition # of matrices

Maple file on Coding and Condition Number and PDF version

Continue via 2.1 clicker questions #7-8 and 10

Go over the last 2 problems on the practice set: 21 and 23:

last col AB is all 0 but B has no col of 0s. [... A.lastcolB]

CA=I. Apply to Ax=0 and reason why A cannot have more columns than rows.

Inverse of a matrix.

twobytwo := Matrix([[a, b], [c, d]]);

MatrixInverse(twobytwo);

three := Matrix([[a, b, c], [d, e, f], [g, h, i]]);

Introduction to Linear maps

scalerow3 := Matrix([[1, 0, 0], [0, 5, 0], [0, 0, 1]]);

scalerow3.three;

swaprows13 := Matrix([[0, 0, 1], [0, 1, 0], [1, 0, 0]]);

swaprows13.three;

usualrowop := Matrix([[1, 0, 0], [0, 1, 0], [-2, 0, 1]]);

usualrowop.three;

The black hole matrix: maps R^2 into the plan but not onto (the range is the 0 vector).

Finish 2.2 and work on 2.3: A matrix has a unique inverse, if it exists. A matrix with an inverse has Ax=b with unique solution x=A^(-1)b, and then the columns span and are l.i...

Repeated methodology: apply inverse, use associativity, use def of inverse to obtain the Identity, use definition of Identity to cancel it.

Matrix algebra

Continue with 2.1 and 2.2. Transpose of a matrix via Wikipedia, including Arthur Cayley. Applications including least squares estimates, such as in linear regression, data given as rows (like Yoda).

Continue via 2.1 clicker questions #9

Continue via 2.1 clicker questions 5-6

Powerpoint file.

Matrix multiplication

Algebra of matrix multiplication: AB and BA...

Chap 1 review clicker questions Review #4: (a: v2 = 4v2. b: v3= -v1 + 2v2 c: v1 + v2 = v3).

End of Material for Test 1

Image 1 Image 2 Image 3 Image 4 Image 5 Image 6 Image 7.

Continue via 2.1 clicker questions 1-4

Problem Set 2 clicker questions

Chap 1 review clicker questions #1-4

Maple Code:

M:=Matrix([[1, 4, 7, 5], [2, 5, 8, 7], [3, 6, 9, 9]]);

ReducedRowEchelonForm(M);

M:=Matrix([[1, 4, 7, 5], [2, 5, 8, 7], [3, 6, 9, 10]]);

ReducedRowEchelonForm(M);

Span1:=Matrix([[1, 4, 7, 5,b1], [2, 5, 8,7,b2], [3, 6, 9,9,b3]]);

Span2:=Matrix([[1, 4, 7, 5,b1], [2, 5, 8,7,b2], [3, 6, 9,10,b3]]);

li1:= Matrix([[1, 4, 7, 5,0], [2, 5, 8,7,0], [3, 6, 9,9,0]]);

li2:= Matrix([[1, 4, 7, 5,0], [2, 5, 8,7,0], [3, 6, 9,10,0]]);

li3:= Matrix([[1, 4, 5,0], [2, 5,7,0], [3, 6,10,0]]);

a1:=spacecurve({[t, 4*t, 7*t, t = 0 .. 1]}, color = red, thickness = 2):

a2:=textplot3d([1, 4, 7, ` vector [1,4,7]`], color = black):

b1:=spacecurve({[2*t,5*t,8*t,t = 0 .. 1]}, color = green, thickness = 2):

b2:=textplot3d([2, 5, 8, ` vector [2,5,8]`], color = black):

c1:=spacecurve({[3*t, 6*t, 9*t, t = 0 .. 1]},color=magenta,thickness = 2):

c2:=textplot3d([3,6,9,`vector[3,6,9]`],color = black):

d1:=spacecurve({[0*t,0*t,0*t,t = 0 .. 1]},color=yellow,thickness = 2):

d2:=textplot3d([0,0,0,` vector [0,0,0]`], color = black):

e1 := spacecurve({[3*t, 6*t, 10*t, t = 0 .. 1]},color=black,thickness = 2):

display(a1, a2, b1, b2, c1, c2, d1, d2,e1);

In R^3, span but not l.i., l.i. but not span, both l.i. and span.

Coffee mixing clicker question

The matrix vector equation and the augmented matrix. Decimals (don't use in Maple) and fractions, and the connection of mixing to span and linear combinations. Geometry of the columns as a plane in R^4, of the rows as 4 lines in R^2 intersecting in the point (40,60)

Take questions. Review linear combination and span. Ax via using weights from x for columns of A versus Ax via dot products of rows of A with x and Ax=b the same (using definition 1 of linear combinations of the columns) as the augmented matrix [A |b]

Finish theorem 4 in 1.4.

1.5: vector parametrization equations of homogeneous and non-homogeneous equations.

definitions Span: represent. Linearly Independent: efficiency. Basis: both.

In R^2, span but not li, li but not span, li plus span.

1.3 clicker questions 1, 2 and 4 and introduce the algebra and geometry of span and linear combinations.

1.4.

1.1 and 1.2 clickers #4 onward

History of linear equations and the term "linear algebra" images, including the Babylonians 2x2 linear equations, the Chinese 3x3 column elimination method over 2000 years ago, Gauss' general method arising from geodesy and least squares methods for celestial computations, and Wilhelm Jordan's contributions.

Gauss quotation. Gauss was also involved in other linear algebra, including the history of vectors, another important "linear" object.

vectors, scalar mult and addition, linear combinations and weights, vector equations and connection to 1.1 and 1.2 systems of equations and augmented matrix

Register the i-clickers

Review vocabulary from day 1 or the hw readings

Elimination

We already saw examples of augmented matrices with 0 solutions, via parallel planes, as well as 3 planes that just don't intersect concurrently:

implicitplot3d({x-2*y+z-2, x+y-2*z-3, (-2)*x+y+z-1}, x = -4 .. 4, y = -4 .. 4, z = -4 .. 4)

implicitplot3d({x+y+z-3, x+y+z-2, x+y+z-1}, x = -4 .. 4, y = -4 .. 4, z = -4 .. 4)

Terminology and Idea: Systems of equations. Unknowns. Coefficients. Solutions. Augmented matrix. Pivots. Algebraic Parametrization. Geometric Plot. Points. Lines. Planes. Gaussian (Echelon) and Gauss-Jordan (ReducedRowEchelon). Homogeneous system. Trivial solution.

1.1 and 1.2 clickers #1-3

Slide

At the same time, begin 1.1 and 1.2 including geometric perspectives, by-hand algebraic Gaussian Elimination and pivots, solutions, plotting and geometry, parametrization and GaussianElimination in Maple. In addition, do #5 with k as an unknown but constant coefficient. Prove using geometry of lines that the number of solutions of a system with 2 equations and 2 unknowns is 0, 1 or infinite.

Look at the geometry using implicitplot3d, number of missing pivots, and parametrization of x+y+z=1.

Algebraic and geometric perspectives in 3-D and solving using by-hand elimination, and ReducedRowEchelon and GaussianElimination.

3 equations 2 unknowns with one solution in the plane R

3 equations 3 unknowns with infinite solutions, one solution and no solutions in R

Mention homework and the class webpages