College of Engineering and Computer Science
Mechanical Engineering Department
Fall 2004 Number 17472 Instructor: Larry Caretto
This week will provide a combination of theoretical and applied topics.
We will first consider the concept of abstract vector spaces. This is a generalization of the basic vector concepts from previous courses. An abstract vector space can handle items multivalued items like vectors, matrices and the set of all the functions sin(n
 x), where n is an integer. We will define an inner product that generalizes the concept of a vector dot product and the norm that generalizes the magnitude of the vector. We will also introduce the concepts of a linearly independent set of vectors, a basis set of vectors that “span the space”, and an orthogonal and orthonormal set of vectors. You should make sure that you understand the concepts presented in the readings on this topic. See pages 33-37 of the notes on matrix analysis for the lecture material.
Next, we will apply these ideas to consider the conditions for a set of simultaneous linear equations to have a solution. There are three possibilities for the equation set, expressed as a matrix equation, Ax = b
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It will have a unique solution only if the rank of the A matrix is the same as the rank of the matrix [ A b ] formed by augmenting the A matrix with one additional column given by b and both ranks equal the number of rows in A .
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If the rank of A equals the rank of [ A b ] but the rank is less than the number of rows in A , we will have an infinite number of solutions.
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If the rank of A does not equal the rank of [ A b ] but there is no solution.
Finally we will discuss methods for solving the equation set, Ax = b . We will present Cramer’s rule as an approach that some people like to use for small sets of equations (three equations or less) and then focus on Gaussian elimination which forms the basis for a series of methods that are used in practical applications for solving Ax = b .
The lecture material for simultaneous linear equations is on pages 8-13 and 15-16 of the notes on matrix analysis. Additional material on Gauss elimination is on pages 2-2 of the notes on numerical solutions of linear equations.
Pages 358 –364 introduce the basic properties of generalized vector spaces, inner product spaces, and norms. There is some additional material on norms on page 903 . We will be using these concepts throughout the course.
Pages 331 –336 discuss the linear independence of vector spaces and its relationship to matrix rank.
Pages 321 –323 introduce the topic of simultaneous linear algebraic equations and the matrix representation of those equations.
The corresponding material on vector spaces is covered on pages 37-41 of the lecture notes on vectors and matrices .
Engineering Building Room 1333
E-mail: lcaretto@csun.edu
Mail Code
8348
Phone: 818.677.6448
Fax: 818.677.7062

Goals and assignments for week two ME501A, L. S. Caretto, Fall 2004 Page 2
Pages 324 –341 discuss the conditions for a system of simultaneous linear equations to have or not to have a solution and the condition for a solution to be unique. In engineering analysis, we always want a unique solution; however, when dealing with complex systems of equations we have to be prepared to handle the cased where we do not obtain a unique solution. Reread pages 331-336 assigned for Tuesday to make sure you understand the important concepts of rank and linear independence are discussed as part of this topic.
Pages 346 –349 discuss the use of Cramer’s rule to solve a system of simultaneous linear equations and show how the concept of rank applies to determinants. Again, determinants are useful for understanding linear systems but should be avoided for computations.
Pages 886 –889 discuss Gaussian elimination, the basic numerical approach for solving systems of simultaneous linear equations.
The corresponding material on simultaneous equations is covered on pages 9-15 and 16-17 of the lecture notes on vectors and matrices . There is an expanded discussion of Gauss elimination on pages 2-9 of the lecture notes on numerical solution of simultaneous linear equations .
Although you should be able to do the problems on this assignment by hand, you can use programs like Excel or MATLAB to check your answers. Download the notes on matrix analysis tools from the Course Notes page on the course web site.
Page 364, problem 3 . This asks a basic question about vector spaces.
Page 365 , problems 19 and 21 . These give you practice in computing vectors norms and inner products as well as determining whether or not two vectors are orthogonal.
Page 330 , problem 15 . This problem on the application of Gaussian elimination to solve a set of equations is an important one. There will be a problem like this on an examination during the semester. You may want to do this first using a software package like Excel or Matlab, but you should be sure that you can do this by hand.
Pages 336-337 , problems 5 , 9 , 15 , and 29 . These examine the evaluation of linear dependence or independence and ask you to apply this concept to determine the rank of matrices.
Pages 350 , problem 19 . This will give you a comparison of using Gaussian elimination and
Cramer’s rule.