Page 42 - Numerical Methods for Chemical Engineering

Existence and uniqueness of solutions 31

(I) the dimensions of K A and R A satisfy the dimension theorem,

dim(K A ) + dim(R A ) = N (1.154)
(II) If K A contains only the null vector 0, dim(K A ) = 0, and so dim(R A ) = N. The range
N
N
therefore completely fills so that every b ∈ is in the range, b ∈ R A , and for any
N
b ∈ , the system Ax = b has a solution.
[2]
[1]
Proof (I) Let us define an orthonormal basis {u , u ,..., u [P] , u [P+1] ,..., u [N] } for N
such that the first P members of the basis span the kernel of A,
[1] [2] [P]
K A = span u , u ,..., u (1.155)
Since the kernel is a subspace and thus satisfies all of the properties of a vector space, we
can always form such a basis. Therefore, we can write any w ∈ K A as

w = c 1 u [1] + c 2 u [2] + ··· + c P u [P] (1.156)
and

dim(K A ) = P (1.157)

N
We now write any arbitrary vector v ∈ as an expansion in this orthonormal basis with
[ j]
the scalar coefficients v j = v · u ,
v = v 1 u [1] +· · · + v P u [P] + v P+1 u [P+1] +· · · + v N u [N] (1.158)

We now operate on this vector by A,
[1] [P] [P+1] [N]
Av = v 1 Au +· · · + v P Au + v P+1 Au +· · · + v N Au
(1.159)
[1] [P] [P+1] [N]
Av = A v 1 u +· · · + v P u + v P+1 Au +· · · + v N Au
As v 1 u [1] +· · · + v P u [P] ∈ K A , this equation becomes

Av = v P+1 Au [P+1] +· · · + v N Au [N] (1.160)
N
Since the range of A is the set of Av for all v ∈ , we see that any vector in the range can
be written as a linear combination of the N − P basis vectors {Au [P+1] ,..., Au [N] }, and
so dim(R A ) = N − P. Thus, dim(K A ) + dim(R A ) = N.
(II) follows directly from (I). QED

What happens to the existence and uniqueness of solutions to Ax = b if K A is not empty?
Let us say that dim(K A ) = P > 0, and that we form the orthonormal basis for A as in the
proof above, such that we can write any w ∈ K A as
[1] [2] [P]
w = c 1 u + c 2 u + ··· + c P u (1.161)
N
We now consider the arbitrary vector v ∈ , written as
v = v 1 u [1] +· · · + v P u [P] + v P+1 u [P+1] +· · · + v N u [N] (1.162)