### Class Highlights

• Wed Jul 6 Final project presentations
• Tues Jul 5
Take questions
Geometry of row operations and determinants
More clicker questions
Why matrix multiplication works the way it does
Matrix theorem
• Fri Jul 1 Test 3
If you are finished early then go to the computer lab and work on the final project [I will bring references to help look up teaching ideas].
Finish the last slide of transformations. Computer graphics demo via definition of triangle := Matrix([[4,4,6,4],[3,9,3,3],[1,1,1,1]]); and then ASULearn Computer Graphics Example D. Also look at Homogeneous 3D coordinates and Example G. Example I.
• Thur Jun 30 Representing linear operators as matrices. Change of basis.
transformations *Note the typo in the rotation matrix
Prove that a rotation matrix rotates algebraically as well as geometrically. Discuss dilation, shear, and reflection. Discuss what Euclidean transformation is missing from our list.
Clicker questions
Clicker questions
• Wed Jun 29 Take questions. Explore why A.P=P.Diag using an example in Maple.
Prove that lambda=0 iff A is not invertible.
Prove that x is and eigenvector iff cx is an eigenvector for c nonzero.
Prove that if there are n distinct eigenvalues then the matrix is diagonalizable.
Give a counterexample to the reverse implication. Go to the computer lab:
Group work
If you are done before we come back together, then work on homework for tomorrow [Schaum's outline chapters 5 and 6]
Revisit linear transformations and prove that differentiation is linear.
• Tues Jun 28
Coffee mixing problem and numerical methods issue related to decimals versus fractions. Algebra and geometry of linear combinations of vectors. Basis for the row space and column space.
Define eigenvalues and eigenvectors [Ax=lambdax, vectors that are scaled on the same line through the origin, matrix multiplication is turned into scalar multiplication]. Prove that the computational methods work and explain why we need the extra equation given by the characteristic polynomial. Prove that high powers are easy to create. Explore the eigenvector decomposition formula:
clicker questions
Dynamical systems demo.
Diagonalizability.
• Mon Jun 27 Historical timeline presentations.
Applications: Traffic and oil pipeline construction. Spanning trees.
group work
Linear Transformations
• Fri Jun 24 Basis for degree less than or equal to 3 polynomials. Basis for 2x2 matrices.
Prove that any orthogonal set of n vectors in Rn must be a basis.
Vector spaces and bases associated to a coefficient matrix: Row space, Column space = Range. Rank. Prove that the dimension row space= dimension of the column space.
Vector space associated to homogeneous solutions: the nullspace. Nullity and relationship to rank.
• Thur Jun 23 Test 2 on vectors, vector spaces, span, l.i. and basis. Discuss the final project. Prove that a basis represents uniquely.
• Wed Jun 22
span and li group work clicker review
• Tues Jun 21
Prove that if cv=0 then c=0 or v=0.
Is the Range or Image of a matrix subspaces of Rn? How about the set of solutions of a matrix equation? The nullspace and nullity.
Prove that if a matrix equation has 2 distinct solutions then it has infinite solutions.
Representations of spaces. Graph paper, span, li, basis and dimension.
• Mon Jun 20 clicker review
a) yes
b) no
Generating subspaces of R2 and R3.
Are the following subspaces? Rotation matrices, det 0 matrices, det non-zero matrices, stochastic matrices, continuous functions, R3 with addition changed to add 1 to each coordinate, continuous functions through (0,1), odd functions, even functions, matrices in Gaussian form.
The vector space axioms allow us to have a notion of flatness or linearity even for abstract items.
• Fri Jun 17
Continue Scalars, vectors and vectors spaces. Prove that the norm of of (u+v) = the norm of u plus the norm of v iff they are in the same direction. Look at subsets of R2 and R3 as candidates for vector spaces.
• Thur Jun 16 Go over test 1. Part 2 of The Growing Importance of Linear Algebra in Undergraduate Mathematics by Alan Tucker.
Scalars, vectors and associativity. Exponentiation. Affine transformations as an operation on R2. Matrix multiplication as an operation on R4.
• Wed June 15 Test 1. Work on the Historical Timeline of People and Discoveries.
Introduction to Linear Algebra reading. Yarn activity.
• Tues June 14 Answer questions. Finish presentations for assignment 1.
• Mon June 13 Presentations for Graded Assignment 1: Definitions and Computational Review
• Fri June 10 Fill out index sheets. Evelyn Boyd Granville worksheet. Introduction to Maple. Linear Regression. Part 1 of the Growing Importance of Linear Algebra in Undergraduate Mathematics by Alan Tucker. Graded Assignment 1: Definitions and Computational Review