Elementary Linear Algebra

Covers the basic problems of linear algebra: solving linear equations, constructing a basis for a vector space, constructing a matrix which represents a linear transformation, eigenvalue-eigenvector and diagonalization.

**Tag(s):**
Linear Algebra

**Publication date**: 31 Dec 2001

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**Views**: 21,111

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**Post time**: 25 Jun 2007 09:41:08

Elementary Linear Algebra

Covers the basic problems of linear algebra: solving linear equations, constructing a basis for a vector space, constructing a matrix which represents a linear transformation, eigenvalue-eigenvector and diagonalization.

Excerpts from the Preface:

Linear algebra, as we know it today, furnishes a beautiful mathematical structure; the integral parts of the structure are powerful tools for engineers, scientists, and mathematicians. Students in these diverse fields come to linear algebra with a common background: They have learned mathematics by working problems and studying examples. With this idea in mind, I have drawn the reader's attention in this text to what I consider to be the five basic problems of linear algebra:

1. The problem of solving linear equations

2. The problem of constructing a basis for a vector space

3. The problem of constructing a matrix which represents a linear transformation

4. The eigenvalue-eigenvector problem

5. The diagonalization problem

These problems give the students landmarks to look for and skills to master in the course.

Of course, the solutions of the five basic problems result in the important theorems of linear algebra. These are stated accurately and then proved unless the proof lies beyond the scope of elementary linear algebra. My hope is that the reader will come to see the structure of linear algebra as supported by the five main beams of the five basic problems with the detail fleshed out by examples, applications, and theorems.

I have given a wide variety of applications of linear algebra. The lion's share are slanted toward science and engineering, however. Presenting applications can be tricky in that enough flavor of the origin of the application must be given to convince the student that the example is not bogus, yet not too much detail of the application must appear to scare away a novice. For example, the presentation of some linear equations arising in structures is the important point, not a minicourse in statics and dynamics. I believe that the applications have been given in appropriate detail and that the students will be able to see and handle the linear algebraic aspects of the applied examples once they are laid out.

Intended Audience:

Except for those sections with the word calculus in the section title, the reader need only be equipped with knowledge of the derivative and integral in an elementary fashion. I have put the deeper connections of calculus and linear algebra in these separate sections so that the interested reader can find them easily.

Linear algebra, as we know it today, furnishes a beautiful mathematical structure; the integral parts of the structure are powerful tools for engineers, scientists, and mathematicians. Students in these diverse fields come to linear algebra with a common background: They have learned mathematics by working problems and studying examples. With this idea in mind, I have drawn the reader's attention in this text to what I consider to be the five basic problems of linear algebra:

1. The problem of solving linear equations

2. The problem of constructing a basis for a vector space

3. The problem of constructing a matrix which represents a linear transformation

4. The eigenvalue-eigenvector problem

5. The diagonalization problem

These problems give the students landmarks to look for and skills to master in the course.

Of course, the solutions of the five basic problems result in the important theorems of linear algebra. These are stated accurately and then proved unless the proof lies beyond the scope of elementary linear algebra. My hope is that the reader will come to see the structure of linear algebra as supported by the five main beams of the five basic problems with the detail fleshed out by examples, applications, and theorems.

I have given a wide variety of applications of linear algebra. The lion's share are slanted toward science and engineering, however. Presenting applications can be tricky in that enough flavor of the origin of the application must be given to convince the student that the example is not bogus, yet not too much detail of the application must appear to scare away a novice. For example, the presentation of some linear equations arising in structures is the important point, not a minicourse in statics and dynamics. I believe that the applications have been given in appropriate detail and that the students will be able to see and handle the linear algebraic aspects of the applied examples once they are laid out.

Intended Audience:

Except for those sections with the word calculus in the section title, the reader need only be equipped with knowledge of the derivative and integral in an elementary fashion. I have put the deeper connections of calculus and linear algebra in these separate sections so that the interested reader can find them easily.

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