In Spring 2018 I'm teaching:
We use WeBWork
for online homework.
Class Notes:Week 1 (1.1, 1.2): We learn how to form the augmented matrix of a system of linear equations. Next we learn about elementary row operations:
-)scale a row by a number,
-)swap two rows,
-)replace a row with itself plus another row.
These operations change the system of linear equations without changing their set of solutions. We use the elementary row operations to put a matrix in Row Echelon Form, and then Reduced Row Echelon Form. The Reduced Row Echelon Form contains information sufficient to determine if the set of solutions is empty, or solve the system if a solution exists.
Homework (1.1; 23, 24, 25) is due 1/17.
Week 2 (1.3, 1.4): We use the reduced echelon form to express the solutions of a matrix equation as the span of a finite set of vectors. The variables associated to pivot columns each have an expression in terms of the free columns.
Homework (1.4; 17-20) is due 1/24, WeBWork is due 1/26.
Week 3 (1.4, 1.5, 1.7): The equation Ax = b has a solution if and only if (iff!) b is in the span of the columns of A, or alternatively iff the augmented column contains no pivots. This means that every vector is in the span of the columns of A iff every row contains a pivot. When a matrix A acts on vectors x,y with scalar c the following is true:
-) A(x + y) = A(x) + A(y)
-) A(cx) = cA(x)
Homework (1.7; 33-38) is due 1/31, WeBWork is due 2/2.
Week 4 (1.7, 1.8, 1.9): Linear transformations are maps T: R^n -> R^m which satisfy the two axioms above: T(x + y) = T(x) + T(y), T(cx) = xT(x). We see that any matrix defines a linear transformation, and conversely, any linear transformation can be represented by a matrix.
Homework(1.9; 23, 24) is due 2/7, WeBWork is due 2/8.
Week 5 (1.9, 2.1, 2.2)
We study the algebra of matrices. Matrix multiplication is associative and distributive, and corresponds to composition of the associated linear transformations. A transformation is invertible if and only if a matrix which represents it also has an inverse. We can compute the inverse of a matrix using elementary row operations.
Homework(2.1; 9, 10) is due 2/14, WeBWork is due 2/16.
Week 6 (2.2, 2.3)
Exam I is next week (2/21)!
Solutions to quizzes:
Week 7 (3.1): We begin working on determinants.
Homework(3.1; 39, 40) is due 2/28, WeBWork is due 3/2.
Week 8 (5.1-5.4): Eigenvectors, eigenvalues, and diagonalization.
Homework(5.1; 1-8) is due 3/7.
Week 9 (5.1-5.4) More Eigenvectors, eigenvalues, and diagonalization.
Given a matrix A, we should be able to carry out the following tasks:
1) Compute the characteristic polynomial of A.
2) Verify that given numbers are eigenvalues of A.
3) For each verified eigenvalue, compute a basis of the corresponding eigenspace.
Week 10 NO LINEAR ALGEBRA HAPPENS BECAUSE OF SPRING BREAK
Week 11 (6.1-6.4) Inner products, length, and orthogonality.
The dot product provides a real vector space with elements of geometry. We can take the length of a vector, compute distances between two vectors, and decide when two vectors are at right angles to each other. This last concept is called orthogonality. Two vectors are orthogonal if their dot product is 0.
Homework(6.1; 24, 28, 29) is due 3/23.
Week 12 (6.4, 6.5) Gram-Schmidt process and Least-Squares problems.
We us the idea of projecting one vector onto another to transform any basis into an orthogonal (and orthonormal if required) basis; this is the Gram-Schmidt procedure.
Homework(6.4; 3,4,7,8) is due 3/30.
Exam I is next week (4/4)!
Solutions to quizzes:
Here are solutions to the practice final exam.