Page 38 - Numerical methods for chemical engineering

24 1 Linear algebra

A
ℜ ℜ
Av
v
x b

A

N
Figure 1.2 Interpretation of a real N × N matrix A as a linear transformation from the domain
N
into the codomain .
N
We have introduced the notation for the set of all real N-dimensional vectors. The term
N
set merely refers to a collection of objects; however, possesses many other properties,
including
closure under addition
N
N
N
if v ∈ and w ∈ , then the vector v + w is also in ;
closure under multiplication by a real scalar
N
N
if v ∈ and c ∈ , then cv ∈ .
N N N
Also, for any u, v, w ∈ ,any c 1 , c 2 ∈ , a null vector 0 ∈ , and for every v ∈
N
defining an additive inverse, −v ∈ , we have the identities
u + (v + w) = (u + v) + w c 1 (v + u) = c 1 v + c 1 u
u + v = v + u (c 1 + c 2 )v = c 1 v + c 2 v
v + 0 = v (c 1 c 2 )v = c 1 (c 2 v) (1.123)
v + (−v) = 0 1v = v
N
N
As these properties hold for , we say that not only constitutes a set, but that it is also
a vector space.
N
Using the concept of the vector space , we now interpret the N × N real matrix A
N
in a new fashion. We note that for any v ∈ , the matrix-vector product with A is also in
N
N
, Av ∈ . This product is formed by the rule
     
a 11 a 12 ... a 1N v 1 a 11 v 1 + a 12 v 2 + ··· + a 1N v N
a 21 a 22 a 2N v 2 a 21 v 1 + a 22 v 2 + ··· + a 2N v N
 ...     

Av =  . .     . 
 . . . . .   .  =  . . 
.
.
.   . 


a N1 a N2 ... a NN v N a N1 v 1 + a N2 v 2 + ··· + a NN v N
(1.124)
N
N
Thus, A maps any vector v ∈ into another vector Av ∈ .
N
Also, since for any v, w ∈ , c ∈ , we have the linearity properties
A(v + w) = Av + Aw A(cv) = cA v (1.125)
N
N
N
we say that A is a linear transformation mapping into itself, A : → . The action
N
of A upon vectors v, w ∈ is sketched in Figure 1.2. From this interpretation of A as