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Course Outline for Linear Algebra

Catalog Description
This course covers matrices, their properties, operations and applications with emphasis on systems of
linear equations; vector spaces, linear independence, and bases of vector spaces; linear transformations,
kernels and ranges; determinants, their properties and applications; eigenvalues and eigenvectors; the
standard inner product on R3; the Gram-Schmidt process; diagonalization of symmetric matrices; and
real quadratic forms.

Course Objectives: See attached.

Prepared by:				Reviewed by:
Prof. Robert Tuskey Dept. of Mathematics				Prof. Linda Padilla Department Chairperson Date
Revised	11/02
Revised	11/98	Revised	02/92
Revised	08/96	Revised	11/91
Revised	10/93	Revised	11/89

Week	Topic or Class Activity
1	Sets, Functions, Matrices
2	More matrices, solving systems of equations
3	Vector spaces
4	More on vector spaces, linear independence
5	Spanning sets, bases, and finite dimensional vector spaces
6	Rank of a matrix, structure of solutions of a system of equations
7	Determinants
8	More on determinants, dot products
9	Orthogonality, The Gram-Schmidt process
10	Linear Transformations and matrix representations
11	Operations on linear transformations, null space and range
12	Change of basis, more on matrix representation
13	Similar matrices, eignevalues and eigenvectors
14	Diagonalization and symmetric matrices
15	Applications

OBJECTIVES
Upon completion of this course you will be able to:

1.	Define "set" and the related terminology.
2.	Define "function" and the related terminology.
3.	Explain what is meant by and be able to form the composition of functions.
4.	Explain what is meant by a system of equations, a solution of the system, a consistent system, an inconsistent system and a homogeneous system of equations.
5.	Define "matrix."
6.	Explain what is meant by an "m x n" matrix, a "square matrix of order n," the "(i,j) entry" of a matrix and the "main diagonal" of a square matrix.
7.	Define and determine the equality of two matrices, the sum of two matrices, the difference of two matrices, the product of two matrices, and the product of the scaler and a matrix.
8.	Use summation notation in the definition of matrix multiplication and proof of certain matrix properties.
9.	Define "transpose of a matrix" and find the transpose of a given matrix.
10.	Make a formal or informal proof of various theorems concerning the above objects and operations.
11.	State all of the algebraic properties of matrix operations as discussed in class.
12.	Prove selected algebraic properties of matrix operations as well as various theorems which are off shoots of these properties.
13.	State what is meant by the zero-matrix, by a diagonal matrix, a scalar matrix, and the identity matrix of order n..
14.	Define "upper triangular form" and "lower triangular form" for a matrix.
15.	Define "singular" matrix, "nonsingular" matrix, and "inverse" of a matrix and find the inverse of a given matrix when it arrives.
16.	Prove various theorems concerning the objects mentioned in Objectives 13 - 15.
17.	Explain the connection between singular and nonsingular matrices to the solution of a system of equations.
18.	Define "n by n" elementary matrices of type I, II, or III.
19.	Prove selected theorems concerning the operation of elementary matrices on a given matrix.
20.	Use elementary matrices to develop a technique for finding the inverse of a given matrix.
21.	Explain what is meant by row-reduced echelon form for a matrix and transform a given matrix into row-reduced echelon form.
22.	Define the three elementary row operations on a matrix.
23.	Explain what is meant by one matrix being row equivalent to a second matrix.
24.	Prove various theorems concerning row equivalence and row-reduced echelon form.
25.	Use matrix techniques discussed in class to solve systems of linear equations.
26.	Define "real vector space" and explain the significance of each of the components of the definition.
27.	Give examples of a vector space.
28.	Define "subspace of a vector space" and give examples.
29.	Determine whether or not a given object is a vector space or subspace.
30.	Use appropriate notation, work problems, and prove selected theorems involving vector spaces and subspaces.
31.	Define "linear combination" of a set of vectors.
32.	State what is meant by a set of vectors "spanning" a vector space.
33.	Explain what is meant by a linearly dependent or linearly independent set of vectors.
34.	Define a "basis" for a vector space.
35.	Explain what is meant by a nonzero vector space.
36.	Define the dimension of a nonzero vector space.
37.	Give examples, use appropriate notation, work problems, and prove selected theorems concerning linear dependence and independence, bases, and dimensions of vector spaces.
38.	Define “row space” and “column space” of an m by n matrix.
39.	Explain what is meant by the row (column) rank of a matrix.
40.	Discuss the structure of a linear system of equations.
41.	Define the “determinant” of an n by n matrix and evaluate the determinant of a given matrix.
42.	Discuss and prove the various properties of determinants and use these properties to aid in solving problems involving determinants.
43.	Define the "minor" of an element a_ijof a matrix A.
44.	Define the "cofactor" of an element a_ijof a matrix A.
45.	Explain and preform the process of finding a determinant by cofactor expansion.
46.	Define the “adjoint” of a matrix A and find the adjoint of a given matrix.
47.	Use appropriate notation and prove selected theorems which demonstrate the connection among the inverse of a matrix, the determinant of a matrix, and the adjoint of a matrix.
48.	Apply determinants in other selected situations as discussed in class.
49.	Define the dot product of two vectors and discuss and/or prove its properties.
50.	State the Cauch-Schwarz inequality and the triangle inequnt by cofactor expansion.
51.	Define the “distance” between two vectors and what are “orthogonal” vectors.
52.	Explain what is meant by an orthogonal set of vectors and an orthonormal set of vectors.
53.	Define and calculate the scalar projection and vector projection of one vector on another.
54.	Use appropriate notation, work problems, and prove selected theorems concerning inner products, the Cauch-Schwarz and triangle inequalities, distance and orthogonality.
55.	Discuss and use the Gram-Schmidt Proality for vectors.
56.	Define "linear transformation" of a vector space V into a vector space W.
57.	State what is meant by the “null space” and “range” of a linear transformation.
58.	Explain what is meant by the matrix representation of a linear transformation.
59.	Find the matrix representation of a given linear transformation.
60.	Define the "sum," "scaler multiple" and "composition" of linear transformations and thereby define a vector space of linear transformations.
61.	State what is meant by a vector space of matrices.
62.	Explain the concept of a coordinate vector with respect to an ordered basis.
63.	Find how coordinate vectors transform under a change of basis.
64.	Define "similar matrices.”
65.	Give examples, use appropriate notation, work problems, and prove selected theorems concerning rank of a matrix, linear transformations, null spaces, ranges, vector spaces of linear transformations, and vector spaces of matrices.
66.	Define "diagonalizable linear transformation" and give example space of matrices.
67.	Define "eigenvalue" and "eigenvector" of a linear transformation, give examples, and find the eigenvalues of eigenvectors of a given matrix.
68.	State what is meant by the characteristic polynomial of a matrix.
69.	Work problems based on the definitions mentioned in objectives 64-68 and theorems based on those definitions.
70.	Explain what is meant by a symmetric matrix and skew symmetric matrix, and determine whether or not a given matrix is symmetric or skew symmetric.
71.	Discuss and/or prove the theorems connecting diagonalization and symmetric matrices.
72.	Define "Real Quadratic Form" and "equivalence" of real quadratic forms.
73.	Explain what is meant by congruent matrices.
74.	Use appropriate notation, work problems and prove selected theorems involving quadratic forms.