Clicker question on interests

April was mathematics awareness month - the theme was 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...

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

Last slide of final research presentations

evaluation

Big picture discussion

Review final research presentations

Clicker survey questions

Review for test 3 and take questions on the study guide, material or final research presentations

Eigenvector comic 1, comic 2

Clicker questions---review of eigenvectors #1-5

Clicker questions--- eigenvector decomposition (5.6) part 2 #1, 2 and 3

Dynamical Systems and Eigenvectors remaining examples

Clicker questions--- eigenvector decomposition (5.6) part 2 #4-6

final research presentations

Hamburger earmuffs and the pickle matrix

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)

Clicker questions in 5.1#1-3

Dynamical Systems and Eigenvectors first example

Clicker questions on eigenvector decomposition (5.6) part 1#1-4 [Solutions: 1. a), 2. c), 3. c), 4. b)]

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

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);

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

>Eigenvectors(Ex4);

>Eigenvectors(Ex3);

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

>Eigenvectors(Ex4);

Clicker questions in Chapter 3 #9

review 2.8 and nullspace

Clicker questions in 2.8

Catalog description:

-Eigenvalues and applications (2.8, 5.1 and 5.6) (after test 3: chap 7 selections)

Begin 5.1: the algebra of eigenvectors and eigenvalues, and connect to geometry and Maple.

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

Application: Foxes and Rabbits

subspace, basis, null space and column space

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.) Two other examples.

Clicker questions in Chapter 3 #3-8

3.3 p. 180-181:

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

Overview of new material for test 2

Clicker questions in Chapter 3 #1 and 2

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]

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 determinator comic

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 - and via elementary row operations - so det A non-zero can be added into Theorem 8 in Chapter 2: What Makes a Matrix Invertible.

Mention 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

application of determinants in geology: volumetric strain

Review linear transformations of the plane, including homogeneous coordinates.

linear transformations comic

Finish linear transformation of the plane Computer graphics demo [2.7]

Clicker questions in 2.7 #3 and 4

Keeping a car on a racetrack

Clicker questions in 2.7 #5-7

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

Clicker questions in 2.7 #8

Linear transformations of 3-space: Computer graphics demo [2.7]

Review What Makes a Matrix Invertible

Clicker questions in 2.7 #1 and 2

Applications of 2.1-2.3:

1.8 (p. 62, 65, & 67-68), 1.9 (p. 70-75), and 2.7

Finish Guess the transformation

Mirror mirror and Sheared Sheap

general geometric transformations on R

In the process, review the unit circle

Computer graphics demo [2.7]

Clicker questions in 2.3 and Hill Cipher # 1 (which is like 2.2 # 21 and 23)

Mention 2.1 #21 and 23 and What Makes a Matrix Invertible

Clicker questions in 2.3 and Hill Cipher # 2 and 3

Linear transformations: Ax=b where A is fixed but b varies: 1 unique solutions, 0 and infinite solutions, and 0 and 1 solutions.

Ax=0 with 1 or infinite solutions, depending on A.

Finish 2.3 via the condition number:

Maple file on Hill Cipher and Condition Number and PDF version

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

Guess the transformation

Test 1 corrections

Clicker questions in 2.1 continued with #7 and 8

multiply comic, identity comic

Clicker questions in 2.2 #1 and 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...

What makes a matrix invertible

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.

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

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 |

Applications of 2.1-2.3:

Hill Cipher history

Maple file on Hill Cipher and Condition Number and PDF version

Multiplicative Inverse for 2x2 matrix:

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

MatrixInverse(twobytwo);

MatrixInverse(twobytwo).twobytwo

simplify(%)

2.2 Algebra: Inverse of a matrix.

Repeated methodology: multiply by the inverse on both sides, reorder by associativity, cancel A by its inverse, then reduce by the identity to simplify:

Applications of multiplication and the inverse (if it exists).

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).

Test 1 review Take review questions for test 1

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

GaussianElimination(s13n15extension);

Clicker questions in 1.3 # 4 and 5

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 and plot:

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

GaussianElimination(M);

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

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

diagonalparallelogram:=spacecurve({[-4*t, -5*t, -1*t, t = 0 .. 1]}, color = black, thickness = 2):

c:=spacecurve({[0, 0, t, t = 0 .. 1]}, color = magenta, thickness = 2):

display(a,b,c,d);

Clicker questions in 1.7 (and l.i. equivalences)

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

Begin Chapter 2:

Continue via Clicker questions in 2.1 1-6

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

Matrix multiplication

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);

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] as opposed to [5,7,9])

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 in 1.3 # 1, 2 and 3.

Clicker question in 1.4

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);

Decimals (don't use in Maple) and fractions. 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).

1.5: vector parametrization equations of homogeneous and non-homogeneous equations. Introduce t*vector1 + vector2 is the collection of vectors that end on the line parallel to vector 1 and through the tip of vector 2

1.7 definition of linearly independent and connection to efficiency of span:

Clicker questions in 1.7 (and l.i. equivalences)

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

Review that c1*vector1 + c2*vector2_on_a_different_line is a plane via:

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

GaussianElimination(span1);

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):

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. The matrix vector equation and the augmented matrix and the connection of mixing to span and linear combinations.

Theorem 4 in 1.4

Clicker questions 1.1 and 1.2 continued with #2 onward:

Parametrize x+y+z=1.

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, Foxtrot vector addition comic by Bill Amend. November 14, 1999. 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).

Clicker questions 1.1 and 1.2 #1. Mention solutions and a glossary on ASULearn.

Prepare to share your name, major(s)/minors/concentrations, and something you learned from hw or class yesterday or had a question on.

Review Gaussian and Gauss-Jordan for 3 equations and 2 unknowns in R

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):

with(plots): with(LinearAlgebra):

Ex1:=Matrix([[1,-2,1,2],[1,1,-2,3],[-2,1,1,1]]);

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)

Ex2:=Matrix([[1,2,3,3],[2,-1,-4,1],[1,1,-1,0]]);

implicitplot3d({x+2*y+3*z=3,2*x-y-4*z=1,x+y-z=0}, x=-4..4,y=-4..4,z=-4..4);

Ex3:=Matrix([[1,2,3,0],[1,2,4,4],[2,4,7,4]]);

implicitplot3d({x+2*y+3*z = 0, x+2*y+4*z = 4, 2*x+4*y+7*z = 4}, x = -13 .. -5, y = -1/4 .. 1/4, z = 3 .. 5, color = yellow)

Ex4:=Matrix([[1,3,4,k],[2,8,9,0],[10,10,10,5],[5,5,5,5]]); GaussianElimination(Ex4);

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

UTAustinXLinearAlgebra.mov 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:

with(LinearAlgebra): with(plots):

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

Do EBG#4

Evelyn Boyd Granville #4

EBG4:=Matrix([[1,1,a],[4,2,b]]);

GaussianElimination(EBG4);

3 equations and 2 unknowns in R

Gaussian and EBG#5 with k as an unknown but constant coefficient:

EBG5:=Matrix([[1,k,0],[k,1,0]]);

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.

Solve the system x+y+z=1 and x+y+z=2 (0 solutions - 2 parallel planes)

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

How to get to the main calendar page: google

Review the following vocabulary, which is also on the ASULearn glossary that I am 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