OPTIMIZATION OF
CHEMICAL PROCESSES
(CHE1011)
Dr. Dharmendra Kumar Bal
Assistant Professor (Sr.)
School of Chemical Engineering
11
Vector
Vector is a directed line segment in N-dimensions.
It has both “length” and “direction”.
Vector in Rn is an ordered set of n real numbers.
v = (1,6,3,4) is in R4
a
a b
T
1 b
column vector: 6
3
4
row vector: (1 6 3 4)
Vector Addition and Vector Subtraction
u1 v1 u1 v1
uv
u 2 v 2 u 2 v 2
u1 v1 u1 v1
u v
u2 v2 u1 v2
The difference of two vectors is the result of adding a negative vector
A – B = A + (-B)
Properties of Vector Addition
Commutative: A+B = B+A
Associative: (A+B)+C = A+(B+C)
There is a ZERO vector 0 = *0, 0, ……., 0+T such that A
+0=0+A=A
Note:
• B + (A-B) = A
• -(-B) = B
• -(A-B) = B-A
Vector Multiplication: By Scalar
αv= α(x1, x2)= (αx1, αx2)
Properties:
1. Distributive: ( A B) A B
( ) A A A
2. Associative: ( ) A ( A)
Linear Combination
Given: vectors v1, v2, …, vn in Rn and n real number
c1, c2, …, cn,
the vector x obtained by
x = c1 v1+ c2 v2+ …+ cn vn
is called a linear combination of v1, v2, …, vn
Examples of linear combination of v1 and v2:
1
v1 5v2
2
0.5v2 0v1 0.5v2
Linear Independence
A set of vectors {v1, v2, …, vk} is called linearly dependent if
there exist scalars c1, c2, …, ck ,
not all zero, such that c1v1 + c2v2 + … + ckvk = 0
The vectors are linearly independent if the above equation is
satisfied ONLY by c1 = c2 = … = ck = 0
Examples:
• The set S = {(1, 2), (2, 4)} is linearly dependent because
−2(1, 2) + 1(2, 4) = (0, 0)
• The set S = ,(1, 0), (0, 1), (−2, 5)- is linearly dependent
because
2(1, 0) −5(0, 1) + 1(−2, 5) = (0, 0)
Linear Independence
Determine whether the following set of vectors is
linearly dependent or linearly
Independent: v1 = (1, 2, 3), v2 = (0, 1, 2), v3 = (−2, 0, 1)
Solution: c1v1 + c2v2 + c3v3 = 0
⇒ c1(1, 2, 3) + c2(0, 1, 2) + c3(−2, 0, 1) = (0, 0, 0)
⇒ (c1−2c3, 2c1+c2, 3c1+2c2 +c3) = (0, 0, 0)
⇒ c1 = c2 = c3 = 0
Therefore, the set of vectors is linearly independent
Linear Independence: Properties
• A set of vectors is linearly dependent if and only if
one of the vectors is a linear combination of the
others.
• Any set of vectors containing the zero vector is
linearly dependent.
• If a set of vectors is linearly independent, then
any subset of these vectors is also linearly
independent.
• If a set of vectors is linearly dependent, then any
larger set, containing this set, is also linearly
dependent.
Dot Product (Inner Product, Scalar
Product) of Vectors
The dot product is a scalar
d
A.B AT B a b c e ad be cf
f
The magnitude of a vector is the dot product of a vector with itself .
A AT A aa bb cc
2
The dot product is related to the angle between two vectors.
A.B A B cos
• Matrix Algebra
• Matrix Operations: Addition
• Matrix Multiplication
• Identity Matrix,
• Zero Matrix,
• Symmetric Matrix
• Diagonal matrix
Matrix Operations: Properties
• Matrix addition:
Matrix Multiplication:
The Determinant of a Square Matrix
• The determinant of a matrix A is denoted by
|A| (or det(A)).
• Determinants exist only for square matrices (n
× n).
Properties of Determinants
• |A|=|A'|.
• If a row or column of A = 0, then |A|= 0.
• If every value in a row or column is multiplied by k, then
|A| = k|A|.
• If two rows or columns are identical, |A| = 0.
• If two rows or columns are linear combination of each
other, |A| = 0
• |A| remains unchanged if each element of a row is
multiplied by a constant and added to any other row.
• |AB| = |A| |B|
• Determinant of a diagonal matrix = product of the
diagonal elements
Inverse of a Matrix
Let A denote the (n × n) matrix.
Let B denote an (n × n) matrix such that AB = BA = I.
If the matrix B exists then A is called invertible. Also B is
called the inverse of A and is denoted by A-1
Let A and B be two matrices whose inverse exists.
Let C = AB. Then the inverse of the matrix C exists and
C-1 = B-1A-1.
Inverse of a Matrix
The inverse (A-1) is defined such that A A-1 is I. Not
every matrix has an inverse.
If no inverse exists, then the matrix is called singular
(non invertible, Det = 0)
If A is non-singular, then A-1 is also non-singular.
If A, B are non-singular, then AB is also non singular
and (AB) -1 = B -1A -1
• (ABC)-1 = C-1B-1A-1
• If A is non-singular, then its transpose is also non-
singular. Also, (AT ) -1 = (A-1)T
Rank of a Matrix
• The rank of a matrix is defined as
rank(A) = the number of linearly independent rows
= the number of linearly independent columns
• If a matrix A of dimension m × n (where n < m) is
of rank n, then A has maximum possible rank and
is said to be of full rank.
• r(O)=0. As long as A is not 0, r(A) > 0.
• In general, the maximum possible rank of an
(m × n) matrix A is min(m, n).
Elementary Row and Column Operations
The following elementary row or column operations yield an
equivalent system:
Interchanges: Two rows (or columns) can be interchanged
Scaling: Multiplying a row (or column) by a nonzero
constant
Sum: The row (or column) can be replaced by the sum of
that row (column) and a nonzero multiple of any other row
(column).
Elementary operations do not change the rank. One can use
ERO and ECO to find the rank of a matrix as follows:
ERO⇒minimum # of rows with at least one nonzero entry
ECO⇒minimum # of columns with at least one nonzero entry
