Classes: Linear Algebra 250

Class 1: Linear Systems April 1, 1997

Introduction to the class.

Class 2: Vectors and Matrices 4/3

Covered 1.1-1.3. Axioms for vectors and matrices. Matrix multiplication is associative.

Class 3: Systems of linear equations 4/8

How to formulate and solve a linear system as a problem in matrix algebra.

Class 4: Vector Subspaces 4/10

Vector subspaces can arise as either a Null Space or by the span of a set of vectors. A spanning set of vectors can be either dependent or independent. One way to think about independence is that there are no redundancies. You get what you pay for. Any vector in the span is a unqiue linear combinaton, which is equivalent to zero being a unique linear combination, i.e. all coefficients must be zero. You can then think of the coefficients as coordinates on the vector subspace.

Class 5: Markov Chains 4/15

1.7, 2.1-2.2. Define Markov Chains as determined by rules for a population moving according to probabilities. A basis for a vector subspace consists of independent vectors. The number of vectors in a basis is the same number as the dimension of the subspace that they span.

Class 6: Rank and Linear Transformations 4/17

2.2-2.3, 3.1. The number of columns of a matrix A is the same as the sum of the rank and the nullity. Another way to see this is that the dimension of the domain space (no. of columns) is the sum of the dimension of the null space and the dimension of its range. A basis for the null space can be extended to a basis of the domain with vectors whose images are independent. We also talked about how the first column of a matrix A is given by Ae₁ where e_i is the ith standard basis vector. This generalizes so that the ith column is Ae_i. At the end we introduced linear transformations, and noticed that the LT's from V to W form a vector space.

Class 7: Abstract Vector Spaces and Ordered Bases 4/22

3.1-3.3. The definition of a vector spaces collects the properties of Euclidean spaces that we have been using. Most important are the scalar multiplication and addition properties. We see the connection between a finite dimensional vector space and Euclidean space by using ordered bases. In fact, using a choice of ordered bases, we can construct an isomorphism with Euclidean space by sending that ordered basis to the standard basis for Euclidean space. Next meeting period is an exam.

Class 8: Determinants 4/29

3.4, 4.1-4.3. A linear transformation between abstract vector spaces V and W can be written in matrix form once ordered bases have been chosen for V and W. A choice of bases, depending on T, can be chosen so that the matrix has all zero entries except for possibly on the diagonal. The number of nonzero entries corresponds to the rank. We also discussed taking determinants and Cramer's rule.

Class: 9 Eigenvalues and Eigenvectors 5/1

5.1. An eigenvalue is a scalar that goes with an eigenvector. Both depend on a matrix or linear transformation from V to V. The linear transformation acts on the eigenvector by multiplication by the eigenvalue scalar. For most matrices, most vectors are not eigenvectors. For a diagonal matrix, the standard basis vectors are eigenvectors. This is an important case. Another important case of eigenvectors is for transition matrices corresponding to a Markov Chain. A steady state vector is an eigen vector with eigenvalue one for the transition matrix.

Class 10: Diagonalization 5/6

5.2 The eigenvector equation Ax=x has a solution when is a root of the characteristic polynomial. This polynomial determines several properties of A. It is given by finding the determinant of A-I. At any rate, the next class is the second midterm covering sections 3.4-6.3. The homework due next week will be put on the web site.