Page 40 - Numerical Methods for Chemical Engineering

Existence and uniqueness of solutions 29

[2]
[1]
[1]
From this orthogonal basis set {w , w ,..., w [N] }, an orthonormal one {u ,
[2]
u ,..., u [N] } may be formed by

u [ j] = w [ j] w [ j] (1.148)
or we can normalize the vectors as we generate them,
k−1 [k]
[k] [k] [ j] [k] [ j] [k] w
w = b − u · b u u = k = 1, 2,..., N (1.149)
w [k]
j=1
While this method is straightforward, for very large N, the propagation of round-off errors is
a problem. We discuss in Chapter 3 the use of eigenvalue analysis to generate an orthonormal
basis set with better error properties.

Subspaces and the span of a set of vectors
[2]
N
[1]
Let us say that we have some set of vectors {b , b ,..., b [P] } in with P ≤ N.We
[1]
[2]
N
then define the span of {b , b ,..., b [P] } as the set of all vectors v ∈ that can be
written as a linear combination of members of the set,
[1] [2] [P] [1] [2] [P]
span b , b ,..., b ≡ v ∈ v = c 1 b + c 2 b + ··· + c P b (1.150)
N
[1]
[2]
We see that span {b , b ,..., b [P] } possesses the properties of closure under addition,
closure under multiplication by a real scalar, and all of the other properties (1.123) required
[2]
N
[1]
to define a vector space. We thus say that span{b , b ,..., b [P] } is a subspace of .
N
Let S be some subspace in . The dimension of S is P, dim(S) = P, if there exists
[1]
[2]
some spanning set {b , b ,..., b [P] } that is linearly independent. If dim(S) = P, then
any spanning set of S with more than P members must be dependent. Since any linearly
N
N
independent spanning set of must contain N vectors, dim( ) = N.
The null space and the existence/uniqueness of solutions
We are now in a position to consider the existence and uniqueness properties of the linear
N
system Ax = b, where x, b ∈ and A is an N × N real matrix. Viewing A as a linear
N
transformation, the problem of solving Ax = b may be envisioned as finding some x ∈
N
that is mapped by A into a specific b ∈ . We now ask the following questions:
N
For particular A and b, when does there exist some x ∈ such that Ax = b? (existence
of solutions)
N
For particular A and b, if a solution x ∈ exists, when is it the only solution? (uniqueness
of solution)
The answers to these questions depend upon the nature of the null space,or kernel,of
N
A, K A , that is defined as the subspace of all vectors w ∈ that are mapped by A into the
null vector, 0 (Figure 1.5). We know that the null space must contain at least the null vector
itself, as for any A,
A0 = 0 (1.151)