Chapter 1. Introduction.
1.1. The Geometry and Algebra of Vectors.
1.1 Problems 1--28
1.2. Length and Angle. The Dot Product. Projections.
1.2 Problems 1--52, 61--67.
1.3. Lines and Planes.
1.3 Problems 1--15, 18--30, 35--38.
In the first two weeks, students should master the operations of vectors including linear combination, and identify all different forms of lines and planes. They should be able to compute the basic calculations, understand the geometric interpretation and apply the ideas to applications, which include how to find the angle between two vectors, how to determine the parallel/perpendicular relation of lines and planes, and how to find the distance from a point to a line/plane by using the orthogonal projection.
2.1. Introduction to Systems of Linear Equations.
2.1 Problems 1--38.
2.2. Direct Methods for Solving Linear Systems.
2.2 Problems 1--18, 23--46.
2.3. Spanning Sets and Linear Independence.
2.3 Problems 1--42.
By the end of the fourth week, students should be able to identify the Row Echelon Form(R.E.F.) and the Reduced Row Echelon Form(R.R.E.F.). They should demonstrate the ability to apply both the Gaussian Elimination and the Gaussian-Jordan Elimination Methods to solve linear systems and express the solution in a vector form. Students should understand the concepts of linear combination, spanning sets and linear dependence/Independence. They should be able to determine whether the given vectors form a spanning set of R^n, whether they are linearly dependent or independent, and find a nontrivial linear dependence relationship if the vectors are linearly dependent.
3.1. Matrix Operations.
3.1 Problems 1--22, 31--36
3.2. Matrix Algebra.
3.2 Problems 1--28.
3.3. The Inverse of a Matrix. Elementary Matrices. The Fundamental Theorem of Invertible Matrices.
3.3 Problems 1--23, 24--40, 48--59.
Elementary Matrices should be covered as time permits. It will not be tested on the final exam.
Midterm Exam 1
3.5. Subspaces, Basis, Dimension, Rank. Coordinates. Rank-Nullity Theorem.
3.5 Problems 1--48, 51, 52.
3.6. Introduction to Linear Transformations.
3.6 Problems 1--25, 29--39.
By the end of Chapter 3, students will be expected to master all the operations of matrices and show a clear understanding of the matrix algebra and the concepts of subspaces, basis. They should be able to find a basis of some subspaces, e.g. row/column/null spaces of a matrix A, to determine the rank and nullity of a matrix, and to change coordinates based on the new basis. They should have the statement of the Rank-Nullity Theorem memorized and demonstrate the ability to apply it, e.g. to find the nullity of A^T. Students should also have a clear understanding about the linear transformation with the associated matrix.
4.1. Introduction to Eigenvalues and Eigenvectors.
4.1 Problems 1--18.
4.2. Determinants: the Cofactor Expansion & the Gaussian Method. Applications: Cramer's Rule & Adjoint.
4.2 Problems 1--52, 57--65.
4.3. Eigenvalues and Eigenvectors of n x n Matrices.
4.3 Problems 1--18.
4.4. Similarity and Diagonalization.
4.4 Problems 1--41.
Students should be able to derive the determinant of a matrix by using different methods and to find all eigenvalues and eigenspaces/basis for each eigenspace of a square matrix. They are expected to understand the concept of the similar matrices and have the ability to determine whether a matrix is diagonalizable, and if so, to find the diagonal matrix with the corresponding inverse matrix.
Midterm Exam 2
5.1. Orthogonality. Orthogonal Matrices.
5.1 Problems 1--21.
5.2. Orthogonal Complements and Orthogonal Projections. The Orthogonal Decomposition.
5.2 Problems 1--22.
5.3. The Gram-Schmidt Process.
5.3 Problems 1--14.
5.4. Orthogonal Diagonalization of Symmetric Matrices.
5.4 Problems 1--12.
Students should be able to use Gram-Schmidt Process to construct an orthogonal basis of a subspace and then to obtain an orthonormal basis if needed. They should show a clear understanding of orthogonal complements and also have the ability to do orthogonal projection and decomposition. From Section 5.4, students are expected to be able to determine if a matrix is orthogonal diagonalizable, but the Orthogonal Diagonalization Process of Symmetric Matrices will not be tested on final exam.