Wed Jul 6 Final project presentations
Tues Jul 5
Geometry of row operations and determinants
More clicker questions
Why matrix multiplication works the way it does
Fri Jul 1
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
Finish the last slide of
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
Discuss dilation, shear, and reflection. Discuss what Euclidean
transformation is missing from our list.
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:
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
Dynamical systems demo.
Mon Jun 27 Historical timeline presentations.
Applications: Traffic and oil pipeline construction. Spanning trees.
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
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
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
Prove that if a matrix equation has 2 distinct solutions then it has
Representations of spaces. Graph paper, span, li, basis and dimension.
Mon Jun 20
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.
Part 1 of the Growing Importance of Linear Algebra in Undergraduate
Mathematics by Alan Tucker.
Graded Assignment 1: Definitions and Computational