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Part B
LINEAR ALGEBRA.
VECTOR CALCULUS
Part B consists of
Chap. 7
Linear Algebra: Matrices, Vectors, Determinants. Linear Systems
Chap. 8
Linear Algebra: Matrix Eigenvalue Problems
Chap. 9
Vector Differential Calculus. Grad, Div, Curl
Chap. 10 Vector Integral Calculus. Integral Theorems.
Hence we have retained the previous subdivision of Part B into four chapters.
Chapter 9 is self-contained and completely independent of Chaps. 7 and 8. Thus, Part B
consists of two large
independent
units, namely, Linear Algebra (Chaps. 7, 8) and Vector
Calculus (Chaps. 9, 10). Chapter 10 depends on Chap. 9, mainly because of the occurrence
of div and curl (defined in Chap. 9) in the Gauss and Stokes theorems in Chap. 10.
CHAPTER 7
Linear Algebra: Matrices, Vectors,
Determinants. Linear Systems
Changes
The order of the material in this chapter and its subdivision into sections has been retained,
but various local changes have been made to increase the usefulness of this chapter for
applications, in particular:
1.
The beginning, which was somewhat slow by modern standards, has been
streamlined, so that the student will see
applications
to linear systems of equations
much earlier.
2.
A reference section (Sec. 7.6) on second- and third-order determinants has been
included for easier access from other parts of the book.
SECTION 7.1. Matrices, Vectors: Addition and Scalar Multiplication,
page 272
Purpose.
Explanation of the basic concepts. Explanation of the two basic matrix
operations. The latter derive their importance from their use in defining vector spaces, a
fact that should perhaps not be mentioned at this early stage. Its systematic discussion
follows in Sec. 7.4, where it will fit nicely into the flow of thoughts and ideas.
Main Content, Important Concepts
Matrix, square matrix, main diagonal
Double subscript notation
Row vector, column vector, transposition
Equality of matrices
Matrix addition
Scalar multiplication (multiplication of a matrix by a scalar)
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Page 145

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Comments on Important Facts
One should emphasize that vectors are always included as special cases of matrices and
that those two operations have properties [formulas (3), (4)] similar to those of operations
for numbers, which is a great practical advantage.
SOLUTIONS TO PROBLEM SET 7.1, page 277
2.
YZ
,
,
,
4.
, same,
, undefined
6.
Undefined, undefined,
, same
8.
Undefined,
, undefined, undefined
10.
1
_
5
A,
_
1
10
A.
Similar (and more important) instances are the scaling of equations in linear
systems, the formation of linear combinations, and the like, as will be shown later.
12.
3, 2,
2
4 and 0, 2, 0. The concept of a main diagonal is restricted to square matrices.
14.
No, no, no. Transposition, which relates row and column vectors, will be discussed
in the next section.
16. (b).
The incidence matrices are as follows, with nodes corresponding to rows and
branches to columns, as in Fig. 152.

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