Scilab Textbook Companion for
Higher Engineering Mathematics
by B. S. Grewal1
Created by
Karan Arora and Kush Garg
B.Tech. (pursuing)
Civil Engineering
Indian Institute of Technology Roorkee
College Teacher
Self
Cross-Checked by
Santosh Kumar, IIT Bombay
Hydraulics and Hydraulic Machines (10CV45)
UNIT 1
DIMENSIONAL ANALYSIS AND MODEL STUDIES
Introduction
Dimensional analysis is a mathematical technique which makes use of the study of
dimensions as an aid to the solution of several engineering problems. It
PEMP
RMD 2501
Introduction to Turbomachines
Session delivered by:
Prof Q.H.
Prof.
Q H Nagpurwala
01
@ M.S.Ramaiah School of Advanced Studies
1
Session Objectives
PEMP
RMD 2501
This session is intended to introduce the following:
Turbomachinery and their
Curves
- Foundation of Free-form Surfaces
Why Not Simply Use a Point Matrix to
Represent a Curve?
Storage issue and limited resolution
Computation and transformation
Difficulties in calculating the intersections or curves
and physical properties of obj
The Newton-Raphson Method
1
Introduction
The Newton-Raphson method, or Newton Method, is a powerful technique
for solving equations numerically. Like so much of the dierential calculus,
it is based on the simple idea of linear approximation. The Newton Me
Lecture 2
The rank of a matrix
Eivind Eriksen
BI Norwegian School of Management
Department of Economics
September 3, 2010
Eivind Eriksen (BI Dept of Economics)
Lecture 2 The rank of a matrix
September 3, 2010
1 / 24
Linear dependence
Linear dependence
To
1.10
Eigenvalues and Eigenvectors
Definition: Let A be an matrix. A scalar is called an
eigenavlue of A if there exists a nonzero vector in such
that = . The vector x is called an eigenvector
corresponding to .
Let us look at the geometrical significance
Problem 6: Find
3
of the matrix 1
1
values.
the
4
2
1
3
Solution: Let = 1
1
sum and product of the eigen values
4
4 without actually finding the eigen
3
4
2
1
4
4
3
Sum of the eigen values = trace of = 3 + 2 + 3 = 4.
Product of the eigen values = .
3
Now,
3
Problem 5: Find the eigen values of when = 5
3
5
Solution: The characteristic equation of
3 94 ) 1 = 0.
Hence the eigen values of are 3, 4, 1.
The eigen values of 5 are 35 , 45 , 15 .
0
4
6
0
0 .
1
is obviously
3
Problem 4: If the given values of = 2
3
1
2
2, 2, 3 find the given values of and .
10
3
5
5
4
7
are
Solution: Since 0 is not an eigen value of , is a non
singular matrix and hence 1 exists.
Eigen values of 1 are
22 , 22 32 .
1 1 1
, ,
2 2 3
and eigen va
Problem 3: Let A be an matrix A with eigenvalues
1 , . . and corresponding eigenvectors 1 , . . . Prove
that if 0, then the eigenvalues of cA are 1 , . . with
corresponding eigenvectors 1 , . . .
Solution: Let be one of the eigenvalues of A with
correspon
Problem 1: Find the eigenvalue and eigenvectors of the
matrix
4
3
=
6
5
Solution: Let us first derive the characteristic polynomial
of A. We get
2 =
4
3
6
1
5
0
0
4
=
1
3
6
5
Note that the matrix 2 is obtained by subtracting
from the diagonal elemen
Problem 2: Find the eigenvalues and eigenvectors of the
5 4 2
matrix 4 5 2
2 2 2
Solution: The matrix 3 is obtained by subtracting
from the diagonal elements of A. Thus
5
3 =
4
2
4
5
2
2
2
2
The characteristic polynomial of A is 3 . Using row
and col
Problem 3: Let A be an matrix A with eigenvalues
1 , . . and corresponding eigenvectors 1 , . . . Prove
that if 0, then the eigenvalues of cA are 1 , . . with
corresponding eigenvectors 1 , . . .
1.9
The Rank of a Matrix
In this module the student is introduced to the concept of
the rank of a matrix. Rank enables one to relate matrices to
vectors, and vice versa. Rank is a unifying tool that enables
us to bring together many of the concepts discus
Problem 3: Find a basis for the subspace of 4 spanned
by the vectors
1,2,3,4 , 1, 1, 4, 2 , 3,4,11,8
Solution: We construct a matrix having these vectors as
row vectors.
1
= 1
3
2
1
4
3
4
4 2
11 8
Determine a reduced echelon form of . We get
1
1
3
2
1
4
Problem 1: Find a basis for the row space of the following
matrix , and determine its rank.
1
= 2
1
2
5
1
3
4
5
Solution: Use elementary row operations to find a reduced
echelon form of the matrix . We get
1
2
1
2
5
1
3
1
4 0
5
0
2
1
1
3
1
2 0
2
0
0
1
0
7
Problem 2: Find a basis for the column space of the
following matrix .
1
= 2
1
1
3
4
0
2
6
Solution: The transpose of is
1
= 1
1
2
3
2
1
4
6
The column space of becomes the row space of . Let us
find a basis for the row space of . Compute a reduced
echel
Exercise
1. Determine the ranks of the following matrices using the
definition of rank.
1 2 1
a. 2 4 2
1 2 3
2 1 3
b. 4 2 6
2 1 3
1 3 4
c. 1 3 1
0 6 5
2. Find the reduced echelon form for each of the following
matrices. Use the echelon form to determine a
1.8
The Inverse of a Matrix
In this module we introduce the concept of the matrix
inverse. We will see how an inverse can be used to solve
certain systems of linear equations, and we will see an
application of matrix inverse in cryptography, the study of
Problem 3: Solve the system of equations
1 2 23 = 1
21 32 53 = 3
1 + 32 + 53 = 2
Solution: This system can be written in the following
matrix form:
1
2
1
1
3
3
1
1
2 = 3
3
2
2
5
5
If the matrix of coefficients is invertible, the unique
solution is
1
1
2 =
Exercise
1. Determine the inverse of each of the following 3 X 3
matrices, if it exists, using the method of Gauss Jordan
elimination.
1 2 3
a. 0 1 2
4 5 3
1 2 3
b. 1 2 1
5 2 3
2. Determine the inverse of each of the following 4 X 4
matrices, if it exists
1.7
Matrix Representations of Linear Transformation
In this module we introduce a way of representing a
linear transformation between general vector spaces by a
matrix. We lead up to this discussion by looking at the
information below that is necessary to
Problem 4: Consider the linear operator , = 2, +
on 2 . Find the matrix of with respect to the standard
basis = 1,0 , 0,1
of 2 . Use the transformation
= 1 to determine the matrix with respect to the
basis = 2,3 , 1, 1 .
Solution: The effect of on the v
Problem 3: Consider the linear transformation : 3 2 ,
defined by , , = ( + , 2). Find the matrix of with
respect to the bases cfw_1 , 2 , 3 and cfw_1 , 2 of 3 and 2 ,
where
1 = 1,1,0 , 2 = 0,1,4 , 3 = 1,2,3 1 = 1,0 , 2 = (0,2).
Use this matrix to find t
Problem 4: Consider the linear operator , = 2, +
on 2 . Find the matrix of with respect to the standard
basis = 1,0 , 0,1
of 2 . Use the transformation
= 1 to determine the matrix with respect to the
basis = 2,3 , 1, 1 .