MA-106 Linear Algebra
Ananthnarayan H.
Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai - 76
12th January, 2015
D2 - Lecture 4
Ananthnarayan H.
D2 - Lecture 4
12th January, 2015
1 / 16
Recall: Elementary and Permutation Matric
MA-106 Linear Algebra
Ananthnarayan H.
Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai - 76
27th January, 2015
D2 - Lecture 10
Ananthnarayan H.
D2 - Lecture 10
27th January, 2015
1 / 15
Recall: A basis for R2
Pick v1 6= 0. Ch
MA-106 Linear Algebra
Ananthnarayan H.
Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai - 76
12th February, 2015
D2 - Lecture 17
Ananthnarayan H.
D2 - Lecture 17
12th February, 2015
1 / 16
Eigenvalues and Eigenvectors: Motivat
MA-106 Linear Algebra
Ananthnarayan H.
Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai - 76
19th January, 2015
D2 - Lecture 7
Ananthnarayan H.
D2 - Lecture 7
19th January, 2015
1 / 13
Summary: N (A) and C(A)
Remark: Let A be
MA-106 Linear Algebra
Ananthnarayan H.
Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai - 76
5th February, 2015
D2 - Lecture 14
Ananthnarayan H.
D2 - Lecture 14
5th February, 2015
1 / 13
Matrix of a Linear Transformation
Let T
MA-106 Linear Algebra
Ananthnarayan H.
Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai - 76
15th January, 2015
D2 - Lecture 6
Ananthnarayan H.
D2 - Lecture 6
15th January, 2015
1/1
Recall: Rank of A
rank(A) = number of column
MA-106 Linear Algebra
Ananthnarayan H.
Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai - 76
22nd January, 2015
D2 - Lecture 9
Ananthnarayan H.
D2 - Lecture 9
22nd January, 2015
1 / 12
Recall: Span and Linear Independence
Let
MA-106 Linear Algebra
Ananthnarayan H.
Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai - 76
16th February, 2015
D2 - Lecture 18
Ananthnarayan H.
D2 - Lecture 18
16th February, 2015
1 / 11
Summary
Let A be n n.
1
The character
MA-106 Linear Algebra
Ananthnarayan H.
Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai - 76
3rd February, 2015
D2 - Lecture 13
Ananthnarayan H.
D2 - Lecture 13
3rd February, 2015
1 / 12
Recall: Gram-Schmidt Process
If the set
MA-106 Linear Algebra
Ananthnarayan H.
Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai - 76
8th January, 2015
D2 - Lecture 3
Ananthnarayan H.
D2 - Lecture 3
Recall
Last time: We saw a system of linear equations, and the
elimi
MA-106 Linear Algebra
Ananthnarayan H.
Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai - 76
10th February, 2015
D2 - Lecture 16
Ananthnarayan H.
D2 - Lecture 16
10th February, 2015
1 / 23
Recall: Determinant - Definition
The
MA-106 Linear Algebra
Ananthnarayan H.
Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai - 76
20th January, 2015
D2 - Lecture 8
Ananthnarayan H.
D2 - Lecture 8
20th January, 2015
1 / 13
Instructions for the Quiz
Short Quiz for
MA-106 Linear Algebra
Ananthnarayan H.
Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai - 76
9th February, 2015
D2 - Lecture 15
Ananthnarayan H.
D2 - Lecture 15
9th February, 2015
1 / 17
Recall: Coordinate Vectors
Let V be a f
MA-106 Linear Algebra
Ananthnarayan H.
Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai - 76
17th February, 2015
D2 - Lecture 19
Ananthnarayan H.
D2 - Lecture 19
17th February, 2015
1 / 14
Summary
Let A be n n.
1
2
A is diagon
MA-106 Linear Algebra
Ananthnarayan H.
Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai - 76
19th February, 2015
D2 - Lecture 20
Ananthnarayan H.
D2 - Lecture 20
19th February, 2015
1 / 10
Recall: Diagonalizability
A matrix A
MA-106 Linear Algebra
Ananthnarayan H.
Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai - 76
29th January, 2015
D2 - Lecture 11
Ananthnarayan H.
D2 - Lecture 11
29th January, 2015
1 / 13
Recall: Orthogonal Subspaces
Definition
MA-106 Linear Algebra
Ananthnarayan H.
Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai - 76
13th January, 2015
D2 - Lecture 5
Ananthnarayan H.
D2 - Lecture 5
13th January, 2015
1/9
Echelon Form
Recall: If A is an n n matrix,
MA-106 Linear Algebra
Ananthnarayan H.
Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai - 76
5th January, 2015
D2 - Lecture 1
Ananthnarayan H.
D2 - Lecture 1
Some Class Policies
Evaluation: 50 marks are waiting to be earned:
T
MA-106 Linear Algebra
Ananthnarayan H.
Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai - 76
2nd February, 2015
D2 - Lecture 12
Ananthnarayan H.
D2 - Lecture 12
2nd February, 2015
1 / 12
Recall
Let W is a subspace of Rn , and
MA-106 Linear Algebra
Ananthnarayan H.
Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai - 76
6th January, 2015
D2 - Lecture 2
Ananthnarayan H.
D2 - Lecture 2
Recall
In the last class, we saw different methods to solve 2 2 and
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Text and references
Main Text:
E. Kreyszig, Advanced Engineering Mathematics, 8th ed.
(Chapters 6 and 7)
Additional references:
1) Notes by Prof. I.K. Rana
2) S. Kumaresan, Linear Algebra- A geometric approach.
Outline of Week-1
1
Matrices
2
Addition, mul
Outline of the week
1
Rank and solvabilty
2
Rn , Subspaces, Linear spans
3
Linear dependence/independence, basis and dimension
4
Row/Column ranks of a matrix, equality
5
Homogeneous/Inhomogeneous linear systems
Rank of matrix
Definition (rank)
be any of
Outline of the week
1
Determinants
2
Invertibilty by determinants
3
Rank by determinants
4
Cramers rule
Invertibility via rank
This topic should have been done last week itself.
Definition (Full rank)
An m n matrix A is said to be of full rank if its rank
Outline of the week
1
Applications of spectral theorem continued
2
Abstract vector spaces
Example 1
Example 1: Find all 2 2 real matrices which are skew-symmetric.
Also find those which are orthogonal among them. Do the same
exercise for 3 3 real matrices
Example 1
1 1
Example i: Let A = 0 1
0 0
2
the inverse is A 3A + 3I .
multiplying.
0
2 . Verify that (A I )3 = [0] and so
1
Compute the same and verify by
0 1 0
0 1 0
0
(A I )2 = 0 0 2 0 0 2 = 0
0 0 0
0 0 0
0
0
3
2
0
(A I ) = (A I )(A I ) =
0
0
0
=
0
0 2
Outline of the week
1
Special matrices
2
Orthogonality of eigenvectors
3
Spectral theorem and diagonalization
Special matrices, complex adjoint
The most basic source of defect in eigenvalues is the existence of
NON ZERO matrices all whose eigenvalues are
Outline of the week
1
Eigenvalue problem
2
Diagonalization
Eigenvalue problem
Eigenvalue Problem (EVP): Let A be an n n i.e. a square matrix.
Find a scalar and a vector v(6= 0) Rn such that Av = v.
Definition (Eigenvalue, eigenvector)
1
The scalar in the
MA 106: Linear Algebra
Autumn 2015
END-TERM EXAMINATION Marking Scheme
1. For a < b, consider the system of equations:
x+
y+
z = 1
ax + by + z =
a2 x + b2 y + 2 z = 2 .
Find the pairs (, ) in terms of a and b for which the system has infinitely many
solu
MA 106 LA
Spring 2013
Name:
Roll No:
Quiz I
06-02-2013
Division:
Tutorial Batch:
1. If xp is a solution of Ax = b and N (A) is the null space of A,
[1+1]
(i) then any vector of the form xp + x0 , where x0 N (A) is also a solution of Ax = b,
() True
(b) Fa