Page 16 - Numerical Methods for Chemical Engineering

Review of scalar, vector, and matrix operations 5

N
We write v ∈ as an expansion in coordinate basis vectors as
[1] [2] [N]
v = v 1 e + v 2 e + ··· + v N e (1.20)
where the components of e [ j] are Kroenecker deltas δ jk ,

 [ j] 
e 1  δ j1 
[ j] 

 e   δ j2  1, if j = k
e [ j] =  .  =  .  δ jk = (1.21)
 2 


.
.
 .   .  0, if j = k
 
[ j] δ jN
e
N
N
N
Addition of two real vectors v ∈ , w ∈ is straightforward,
     
v 1 w 1 v 1 + w 1
v 2 w 2 v 2 + w 2
     
     
. (1.22)
v + w =  .  +  .  =  . 
. . .
 .   .   
v N w N v N + w N
N
as is multiplication of a vector v ∈ by a real scalar c ∈ ,
   
v 1 cv 1
 v 2   cv 2 
(1.23)
   
cv = c  .  =  . 
.
.
 .   . 
v N cv N
N
For all u, v, w ∈ and all c 1 , c 2 ∈ ,
u + (v + w) = (u + v) + w c(v + u) = cv + cu
u + v = v + u (c 1 + c 2 )v = c 1 v + c 2 v (1.24)
v + 0 = v (c 1 c 2 )v = c 1 (c 2 v)
v + (−v) = 0 1v = v
N
where the null vector 0 ∈ is
0
 
0
 
(1.25)
 
0 =  . 
.
 . 
0
N
We further add to the list of operations associated with the vectors v, w ∈ the dot
(inner, scalar) product,
N

v · w = v 1 w 1 + v 2 w 2 +· · · + v N w N = v k w k (1.26)
k=1