Download Vector spaces and solution of simultaneous equations

College of Engineering and Computer Science
Mechanical Engineering Department
Mechanical Engineering 501A
Seminar in Engineering Analysis
Fall 2004 Number 17472 Instructor: Larry Caretto
Goals and Assignments for the Second Week
Goals for week two
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

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.

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.

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.
Reading for August 31, 2004
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
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Goals and assignments for week two
ME501A, L. S. Caretto, Fall 2004
Page 2
Reading for September 2, 2004
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.
Homework due September 7, 2004
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.