Basic Math Operations - Mathematics and Statistics for the Environment - page 73

2. Convert psi to psf:

70 psi × 14 4 in. 2 /ft 2 = 10,080 psf

■ EXAMPLE 2.67
Problem: The pressure in a pipeline is 6476 psf. What is the head on the pipe?

Solution:

Head on pipe = Feet of pressure

Pressure = Weight × height

6476 psf = 62.4 lb/ft 3 × height

Height = 104 ft

2.13 REVIEW OF ADVANCED ALGEBRA KEY TERMS/CONCEPTS
Advanced algebraic operations (linear, linear differential, and ordinary differential equations) have
in recent years become an essential part of the mathematical background required by environmen-
tal engineers, among others. It is not the intent here to provide complete coverage of the topics
(environmental practitioners are normally well grounded in these critical foundational areas), but it
is important to review the key terms and relevant concepts. Key definitions include the following:

Algebraic multiplicity of an eigenvalue —The algebraic multiplicity of eigenvalue c of matrix
A is the number of times the factor ( t - c ) occurs in the characteristic polynomial of A .
Basis for a subspace —A basis for subspace W is a set of vectors { v 1 , …, v k } in W such that

1. { v 1 , …, v k } is linearly independent, and

2. { v 1 , …, v k } spans W.
Characteristic polynomial of a matrix —The characteristic polynomial of n × n matrix A is the
polynomial in t given by the formula det( A - tI ).
Column space of a matrix —The subspace spanned by the columns of the matrix considered
as a set of vectors (also see row space).
Consistent linear system —A system of linear equations is consistent if it has at least one
solution.
Defective matrix —Matrix A is defective if A has an eigenvalue whose geometric multiplicity
is less than its algebraic multiplicity.
Diagonalizable matrix —A matrix is diagonalizable if it is similar to a diagonal matrix.
Dimension of a subspace —The dimension of subspace W is the number of vectors in any
basis of W . (If W is the subspace { 0 }, then we say that its dimension is 0.)
Echelon form of a matrix —A matrix is in row echelon form if

1. All rows that consist entirely of zeros are grouped together at the bottom of the matrix,

and

2. The first (counting left to right) nonzero entry in each nonzero row appears in a column
to the right of the first nonzero entry in the preceding row (if there is a preceding row).
Eigenspace of a matrix —The eigenspace associated with the eigenvalue c of matrix A is the
null space of A - cl.
Eigenvalue of a matrix —An eigenvalue of matrix A is scalar c such that A x = c x holds for
some nonzero vector x .
Eigenvector of a matrix —An eigenvector of square matrix A is a nonzero vector x such that
A x = c x holds for some scalar c .
Elementary matrix —A matrix that is obtained by performing an elementary row operation
on an identity matrix.