Course Outline for Linear Algebra
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
|Week||Topic or Class Activity|
|1||Sets, Functions, Matrices|
|2||More matrices, solving systems of equations|
|4||More on vector spaces, linear independence|
|5||Spanning sets, bases, and finite dimensional
|6||Rank of a matrix, structure of solutions of a
|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
|12||Change of basis, more on matrix representation|
|13||Similar matrices, eignevalues and eigenvectors|
|14||Diagonalization and symmetric matrices|
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.
|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
|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
|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
|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
|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 aijof a matrix A.|
|44.||Define the "cofactor" of an element aijof 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
|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.|