MA 106 - Linear Algebra
Neela Nataraj
Department of Mathematics,
Indian Institute of Technology Bombay,
Powai, Mumbai 76
neela@math.iitb.ac.in
January 31, 2013
Neela Nataraj
Lecture 8 : D3
Outline of the lecture
Basis & properties
More Subspaces .
Example

Treatment and disposal of Hazardous
Wastes
Treatment methods:
Physical Processes
Chemical Processes
Biological Processes
About 4.4% of hazardous waste generated in the country
is of the nature, which has to be incinerated.
Besides,
segregated
organic
re

HAZARDOUS WASTE
MANAGEMENT
ES 624
Munish K. Chandel
CESE
Quiz 1
What is Hazardous Waste?
An example of hazardous waste disaster.
Briefly explain.
What are your expectations from the course?
What should be the evaluation criteria for this
course?
Sylla

MA 106 - Linear Algebra
Neela Nataraj
Department of Mathematics,
Indian Institute of Technology Bombay,
Powai, Mumbai 76
neela@math.iitb.ac.in
January 28, 2013
Neela Nataraj
Lecture 7 : D3
Outline of the lecture
Vector spaces, Subspaces
Examples
Cardinali

Hazardous waste Management in
the United States
Resource Conservation and Recovery Act
The Resource Conservation and Recovery Act commonly
referred to as RCRA is US primary law governing the
disposal of solid and hazardous waste.
Congress passed RCRA on

MA 106 Linear Algebra (Spring 2016)
1
Linear Maps and Matrix Representation
1. Let f : Rn Rm be a linear map. Show that f (0) = 0, i.e., f must map the
zero vector in Rn to the zero vector in Rm . Further show that f must map lines
in Rn passing through t

MA 106 Linear Algebra (Spring 2016)
Tut Sheet 3: Determinants, Vector Spaces, Subspaces and Bases
1. Adopt the Gaussian Elimination Method (GEM) to obtain an algorithm to determine
the determinant of a square matrix A.
2. Compute the inverse of the follow

MA 106 Linear Algebra (Spring 2016)
Tut Sheet 6: Eigenvalues and Eigenvectors
1. If the n n matrices A and A are similar (i.e. A = P 1 AP for some nonsingular
n n matrix P ) then show that
(i) A and A have same eigenvalues;
(ii) if v is an eigenvector of

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MA 106 Linear Algebra (Spring 2016)
Tut Sheet 5: Rank of a Matrix
1. Let A be a m n matrix and let fA be the linear transformation from Rn Rm
induced by A. Show that column-rank (A) = dim (Range (fA )=rk fA .
2. Prove that under elementary row operations,

MA 106 Linear Algebra (Spring 2016)
1
Linear Maps and Matrix Representation
1. Let f : Rn Rm be a linear map. Show that f (0) = 0, i.e., f must map the
zero vector in Rn to the zero vector in Rm . Further show that f must map lines
in Rn passing through t

MA 106 Linear Algebra (Spring 2016)
Tut Sheet 7: Inner Product Spaces, Special Matrices
and Quadratic Forms
1. On the vector space C 1 [a, b] of continuously differentiable real valued functions,
examine whether or not hf, gi, defined below is an inner pr

Operations on matrices
Set of all matrices of size m n is denoted by Mm,n
As before, for A = (aij ), B = (bij ) Mm,n and R,
A + B := (aij + bij );
A := (aij )
Matrix addition and scalar multiplicaion satisfies the
usual properties. The zero element is the

MA 106 Linear Algebra (Spring 2016)
Tut Sheet 4: Linear Transformations of Vector Spaces
Note: Problems in the previous tutorial sheet pertaining to vector spaces, subspaces and bases (Q. 719)
may be discussed in this tutorial if they have not been covere

MA 106 - Linear Algebra
Neela Nataraj
Department of Mathematics,
Indian Institute of Technology Bombay,
Powai, Mumbai 76
neela@math.iitb.ac.in
January 17, 2013
Neela Nataraj
Lecture 4: D3
Outline of the lecture
Vector spaces in Rn
Vector Subspaces Tut. Sh

Environmental Audit identify
environmental problems at the origin of
waste production
> Practices that reduce or eliminate
the hazardous waste
Pollution Prevention
and Minimization
Pollution Prevention
Though an obvious concept, this has been one
of the

TOXICOLOGY
TOXICOLOGY
A waste is hazardous when exhibits any of
the following characteristics:
Ignitable
Corrosive
Reactive
Toxic
TOXICOLOGY
Fundamental objective of HWM is the protection of human health by reducing the
risk, if not the toxicity, of

MA 106 - Linear Algebra
Neela Nataraj
Department of Mathematics,
Indian Institute of Technology Bombay,
Powai, Mumbai 76
neela@math.iitb.ac.in
February 7, 2013
Neela Nataraj
Lecture 10 : D3
Outline of the lecture
Matrix of a linear transformation Tut. She

MA 106 - Linear Algebra
Neela Nataraj
Department of Mathematics,
Indian Institute of Technology Bombay,
Powai, Mumbai 76
neela@math.iitb.ac.in
January 7, 2013
Neela Nataraj
Lecture 1
Outline of the lecture
Matrices (A revision) & Gauss Elimination
Matrix

MA 108 - Ordinary Differential Equations
Neela Nataraj
Department of Mathematics,
Indian Institute of Technology Bombay,
Powai, Mumbai 76
neela@math.iitb.ac.in
April 4, 2013
Neela Nataraj
Lectures 11 : D3
Outline of the lecture
Annihilator Method- Recap
L

Fate and Transport of
Contaminants
The release of the contaminants into the
environment is inevitable
Contaminants are released through:
Manufacture and use of products
Result of treatment and disposal of the waste
Could move quickly or slow to the r

MA 106 - Linear Algebra
Neela Nataraj
Department of Mathematics,
Indian Institute of Technology Bombay,
Powai, Mumbai 76
neela@math.iitb.ac.in
January 24, 2013
Neela Nataraj
Lecture 6: D3
Outline of the lecture
Row space, Column space : Examples
Row Rank

MA 108 - Ordinary Differential Equations
Neela Nataraj
Department of Mathematics,
Indian Institute of Technology Bombay,
Powai, Mumbai 76
neela@math.iitb.ac.in
April 15, 2013
Neela Nataraj
Lectures 13 : D3
Outline of the lecture
Properties of Laplace tran

Current Hazardous Waste
Management Practices
Environmental audit
Pollution prevention
Facility development and operations
Environmental Audit
Environmental Audit
Environmental Audit is a systematic, documented, periodic and
objective process in assessing

MA 106 - Linear Algebra
Neela Nataraj
Department of Mathematics,
Indian Institute of Technology Bombay,
Powai, Mumbai 76
neela@math.iitb.ac.in
January 10, 2013
Neela Nataraj
Lecture 2
Outline of the lecture
Gauss Elimination Method
Row-Echelon Form, Row R

MA 108 - Ordinary Differential Equations
Neela Nataraj
Department of Mathematics,
Indian Institute of Technology Bombay,
Powai, Mumbai 76
neela@math.iitb.ac.in
April 8, 2013
Neela Nataraj
Lectures 12 : D3
Outline of the lecture
Laplace Transforms-Recap
Pr

MA 106 - Linear Algebra
Neela Nataraj
Department of Mathematics,
Indian Institute of Technology Bombay,
Powai, Mumbai 76
neela@math.iitb.ac.in
February 4, 2013
Neela Nataraj
Lecture 9 : D3
Outline of the lecture
Cramers Rule
Linear transformations
Example