clicker. Share the final research presentations topic with the rest of the class (name, major(s), concentrations/minors, research project idea, and whether you prefer to go 1st, 2nd or have no preference).

informal eval while I check in about the projects, and then formal evaluations.

Take questions on the final research presentations

Hamburger earmuffs and the pickle matrix

clicker.

April is mathematics awareness month - the theme is magic, mystery and mathematics. Have the class give me a 3x3 matrix. Look at

h,P:=Eigenvectors(A)

MatrixInverse(P).A.P

which (ta da) has the eigenvalues on the diagonal (when the columns of P form a basis for R

Applications to mathematical physics, quantum chemistry...

big picture discussion

clicker survey questions

Discuss the final research presentations.

eigenvector clicker review

3.3 and 2.8 clickers #6-8

Finish Dynamical Systems and Eigenvectors

Review: Ax=b solutions when mult makes sense, when b=0, when A is invertible, when A=Matrix([[1,0],[0,1],[0,0]]) (and column space and null space). Elementary matrix.

eigenvector decomposition clickers 2 #3-5

Review eigenvectors and eigenvalues:

definition (algebra and geometry)

What equations have we seen

Why we use det(A-lambdaI)=0

Why we use the eigenvector decomposition versus high powers of A for longterm behavior (reliability)

Continue Dynamical Systems and Eigenvectors

Highlight predator prey, predator predator or cooperative systems (where cooperation leads to sustainability)

eigenvector decomposition clickers 2 #1 and 2

Dynamical Systems and Eigenvectors first example

eigenvector decomposition clickers 1

5.1 clicker questions

Finish Geometry of Eigenvectors and compare with Maple

>Ex4:=Matrix([[1/2,1/2],[1/2,1/2]]);

>Eigenvectors(Ex4);

Begin 5.6: Eigenvector decomposition for a diagonalizable matrix A_nxn [where the eigenvectors form a basis for all of R

Foxes and Rabbits

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____].

Continue 5.1: Review the algebra of eigenvectors and eigenvalues. [Ax=lambdax, vectors that are scaled on the same line through the origin, matrix multiplication is turned into scalar multiplication]. Solving Ax=lambdax algebraically using determinant(A-lambdaI)x=0, and substituting each lambda in to find a basis for the eigenspaces of A and equivalently the nullspace of (A-lambda I).

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

Geometry of Eigenvectors and compare with Maple

>Ex1:=Matrix([[0,1],[1,0]]);

>Eigenvalues(Ex1);

>Eigenvectors(Ex1);

>Ex2:=Matrix([[0,1],[-1,0]]);

>Eigenvectors(Ex2);

>Ex3:=Matrix([[-1,0],[0,-1]]);

>Eigenvectors(Ex3);

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

Algebra: Show that we can solve Ax=lambdax using det(A-lambdaI)=0 and (A-lambdaI)x=0 (ie the nullspace of A-lambdaI).

Eigenvectors of Matrix([[0,0],[1,0]]); and the Eigenvectors command in Maple

Review the LaTex Beamer slide

The relationship of row operations to the geometry of determinants - row operations can be seen as shear matrices when written as elementary matrix form, which preserve area, volume, etc...

Clicker questions #1-3

Determinants 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]

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

Determinant(M); MatrixInverse(M);

M:=Matrix([[a,b,c,d],[e,f,g,h],[i,j,k,l],[m,n,o,p]]);

Determinant(M); MatrixInverse(M);

LaTex Beamer slides

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

The determinant of A transpose and A triangular (such as in Gaussian form).

The determinant of A inverse via the determinant of the product of A and A inverse - so det A non-zero can be added into Theorem 8 in Chapter 2.

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

Clicker review of race track transformations

Begin Yoda (via the file yoda2.mw) with data from Kecskemeti B. Zoltan (Lucasfilm LTD) as on Tim's page

2.3 clicker review

Clicker review of linear transformations

Review of linear transformations of the plane, including homogeneous coordinates and the extension:

Keeping a car on a racetrack

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

Clicker review of linear transformations

Finish general geometric transformations on R

Computer graphics demo [2.7]

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

Guess the transformation

general geometric transformations on R

In the process, review the unit circle

Clicker questions and review the Hill Cipher

Counterexamples for false statements [If A then B counterexample: A is true but the conclusion B is false]

Can a matrix equation have both 1 and infinite solutions but never be inconsistent?

Ax=0 where A varies

Ax=b where A is fixed but b varies

Maple file on Hill Cipher and Condition Number and PDF version

Condition # of matrices

Review guidelines for Problem Sets, including

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

Dilation inverses

Hill Cipher: Linear transformation of uncoded message vectors to coded message vectors.

A.[uncoded vector] = [coded vector]

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 |

Maple file on Hill Cipher and Condition Number and PDF version

Review Theorem 8 in 2.3 [without linear transformations] for square matrices via clicker question. Discuss what it means for a square matrix that violates one of the statements. Discuss what it means for a matrix that is not square (all bets are off) via counterexamples.

Catalog description:

-2.1-2.3 Applications: Hill Cipher, 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... as time allows)

-Final research sessions

Applications: Introduction to Linear Maps

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

Dilation by 2 matrix

Linear transformations in the cipher setting:

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 |

Answer clicker question #1

Obtain A inverse via elementary row operations E_p...E_2.E_1.A = I and Gaussian reductions of [A|I]

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

MatrixInverse(three);

scalerow2 := Matrix([[1, 0, 0], [0, 5, 0], [0, 0, 1]]);

scalerow2.three;

swaprows12 := Matrix([[0, 1, 0], [1, 0, 0], [0, 0, 1]]);

swaprows12.three;

usualrowop := Matrix([[1, 0, 0], [0, 1, 0], [-4, 0, 1]]);

usualrowop.three;

The rest of the clicker questions for 2.2

Theorem 8 in 2.3 [without linear transformations]: 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:

Inverse of a matrix.

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

MatrixInverse(twobytwo);

MatrixInverse(twobytwo).twobytwo

simplify(%)

Clicker question on glossary

Test 1 corrections

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

MatrixInverse(three);

scalerow2 := Matrix([[1, 0, 0], [0, 5, 0], [0, 0, 1]]);

scalerow2.three;

Obtain via elementary row operations and Gaussian reductions of [A|I]

Take questions on the study guide.

clicker review of past hw questions

spans but not l.i, li but doesn't span, both

1.7 clicker questions - since we didn't finish them in class, here are solutions

Review matrix multiplication and matrix algebra. Introduce 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).

2.1 clicker questions # 7-9

Begin Chapter 2:

Continue via 2.1 clicker questions 1-5

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

Matrix multiplication Algebra of matrix multiplication: AB and BA...

Take questions on 1.7. Review Maple from Wednesday

Linear Combination check of adding a vector that is outside the plane containing Vector([1,2,3]), Vector([4,5,6]), Vector([7,8,9]), ie b3+b1-2*b2 not equal to 0: Vector([5,7,10])

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

ReducedRowEchelonForm(M);

Span check with additional vector:

span2:=Matrix([[1, 4, 7, 5,b1], [2, 5, 8,7,b2], [3, 6, 9,10,b3]]);

GaussianElimination(span2);

Linearly independent check with additional vector:

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

Removing Redundancy

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

Adding the additional vector to the plot:

e1:=spacecurve({[5*t,7*t,10*t,t = 0 .. 1]},color=black,thickness = 2):

e2:=textplot3d([5,7,10,` vector [5,7,10]`], color = black):

display(a1, a2, b1, b2, c1, c2, d1, d2,e1,e2);

Clicker questions:

1.7 clicker questions # 1, 2 and 6

Take questions. Review the geometry of v1+tv2

Review 1.4 #31 and 33

1.7 definition of linearly independent and connection to efficiency of span

l.i. equivalences and clicker

In R^2: spans R^2 but not li, li but does not span R^2, li plus spans R^2.

Examples in R^3 via Maple Code:

Linearly independent and span checks:

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

ReducedRowEchelonForm(li1);

span1:=Matrix([[1, 4, 7, b1], [2, 5, 8,b2], [3, 6, 9,b3]]);

GaussianElimination(span1);

Plotting - to check whether they are in the same plane:

a1:=spacecurve({[t, 2*t, 3*t, t = 0 .. 1]}, color = red, thickness = 2):

a2:=textplot3d([1, 2, 3, ` vector [1,2,3]`], color = black):

b1:=spacecurve({[4*t,5*t,6*t,t = 0 .. 1]}, color = green, thickness = 2):

b2:=textplot3d([4, 5, 6, ` vector [4,5,6]`], color = black):

c1:=spacecurve({[7*t, 8*t, 9*t, t = 0 .. 1]},color=magenta,thickness = 2):

c2:=textplot3d([7,8,9,`vector[7,8,9]`],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):

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

Collect hw and take questions. Review span of the columns (1.3 and 1.4) compared to the span of the solutions of a system of equations (1.5) via examples:

The algebra and geometry of a system of equations with solutions a plane in R^5 off the origin.

s13n15extension:=Matrix([[1,-5,b1],[3,-8,b2],[-1,2,b3]]);

Clicker question. Then discuss what happens when we correctly use GaussianElimination(s13n15extension) - write out the equation of the plane that the vectors span. Choose a vector that violates this equation to span all of R^3 instead of the plane:

M:=Matrix([[1,-5,0,b1],[3,-8,0,b2],[-1,2,1,b3]]);

Theorem 4 in 1.4.

Review that t*vector1 + vector2 is the collection of vectors that end on the line parallel to vector 1 and through the tip of vector 2

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). Maple commands:

Coff:=Matrix([[.3,.4,36],[.2,.3,26],[.2,.2,20],[.3,.1,18]]);

ReducedRowEchelonForm(Coff);

Coffraction:=Matrix([[3/10,4/10,36],[2/10,3/10,26],[2/10,2/10,20],[3/10,1/10,18]]);

ReducedRowEchelonForm(Coffraction);

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

1.3 clicker questions #4 and continue the algebra and geometry of span and linear combinations.

Begin 1.4. 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].

The matrix vector equation and the augmented matrix.

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

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. linear combination language (addition and scalar multiplication of vectors). #8 in multiple choice questions from 1.1 and 1.2

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.

Take questions on the glossary / syllabus. clicker questions.

Finish 1.2 examples and review Gaussian/Gauss-Jordan, Maple and geometry.

T/F: A linear system of 3 equations and 3 unknowns, where no 2 of the equations are multiples, can be inconsistent...

Reminder: You'll need your clickers on Wednesday.

Take a look at the number of solutions, the algebra and geometry arising from:

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

Review the following vocabulary, which is also on the ASULearn glossary that Dr. Sarah is experimenting with.

augmented matrix

coefficients

consistent

free

Gaussian elimination / row echelon form (in Maple GaussianElimination(M))

Gauss-Jordan elimination / reduced row echelon form (in Maple ReducedRowEchelonForm(M))

homogeneous system

implicitplot

implicitplot3d

linear system

line

parametrization

pivots

plane

row operations / elementary row operations

solutions

system of linear equations

unique

hw is on the calendar page

Collect the hw due at the beginning of class and pass around the attendance sheet.

Ask for any questions on the hw. Go over the algebra (Gaussian) and geometry of 21 if it wasn't already asked about.

Mention that solutions are on ASULearn and are part of the hw for Fri

Please remind how to get to the main calendar page: google

Give the 2 handouts to those not there on Monday.

Gauss-Jordan elimination on 3 equations in 2 unknowns.

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

Gaussian and Gauss-Jordan or reduced row echelon form in general: section 1.2, focusing on algebraic and geometric perspectives and solving using by-hand elimination of systems of equations with 3 unknowns. Follow up with Maple commands and visualization: ReducedRowEchelon and GaussianElimination as well as implicitplot3d in Maple (like on the handout):

Highlight:

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

History of solving equations 1.1 Work on the introduction to linear algebra handout motivated from Evelyn Boyd Granville's favorite problem (#1-3). At the same time, begin 1.1 (and some of the words in 1.2) including geometric perspectives, by-hand algebraic Gaussian Elimination and pivots, solutions, plotting and geometry, parametrization and GaussianElimination in Maple for systems with 2 unknowns in R

Evelyn Boyd Granville #3:

implicitplot({x+y=17, 4*x+2*y=48},x=-10..10, y = 0..40);

implicitplot({x+y-17, 4*x+2*y-48},x=-10..10, y = 0..40);

EBG3:=Matrix([[1,1,17],[4,2,48]]);

GaussianElimination(EBG3);

ReducedRowEchelonForm(EBG3);

Course intro slides

Gaussian

In addition, do #4 and #5 with k as an unknown but constant coefficient.

Evelyn Boyd Granville #4

GaussianElimination(EBG4);

Evelyn Boyd Granville #5

GaussianElimination(EBG5);

ReducedRowEchelonForm(EBG5);

Prove using geometry of lines that the number of solutions of a system with 2 equations and 2 unknowns is 0, 1 or infinite.

Mention homework and the class webpages

pointplot

spacecurve