MH1201 - LINEAR ALGEBRA II
AY 2014-2015
Dr. Le Hai Khoi ([email protected])
Division of Mathematical Sciences, SPMS, NTU

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Preface
These lecture notes are to be used together with the textbook [1] and the reference-book
[2]. During the lectures we will follow these notes carefully. However, you might find these
notes to be too brief and lacking in examples. The textbooks on the other hand have plenty
of examples and exercises, so when you find these notes lacking, you can try to look up the
corresponding material in the textbooks.
For several theorems and their proofs, we have
benefited from some different sources other than the textbooks.
You should also know that these notes will be updated continuously during the course
as we will surely find errors along the way. We would be very thankful if you could report
any mistakes you find to us so they can be corrected as quickly as possible.
We are now ready to start investigating the exciting topic of Linear Algebra II.
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Contents
Contents
1
1
Vector Spaces
3
1.1
Basic Notation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2
Subspaces
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.3
Linear Combinations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.4
Linear Dependence and Independence
. . . . . . . . . . . . . . . . . . . . . .
9
1.5
Bases and Dimension
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.6
Some Applications
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2
Linear Transformations
19
2.1
Basic Notation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.2
The Matrix Representation of a Linear Transformation
. . . . . . . . . . . .
25
2.3
Composition of Linear Transformations
. . . . . . . . . . . . . . . . . . . . .
28
2.4
Invertibility and Isomorphisms
. . . . . . . . . . . . . . . . . . . . . . . . . .
31
2.5
The Change of Coordinate Matrix
. . . . . . . . . . . . . . . . . . . . . . . .
34
3
Diagonalization
37
3.1
Eigenvalues and Eigenvectors
. . . . . . . . . . . . . . . . . . . . . . . . . .
37
3.2
Diagonalizability
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
4
Inner Product Spaces
47
4.1
Inner Products and Norms
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
4.2
The Gram-Schmidt Orthogonalization Process
. . . . . . . . . . . . . . . . .
49
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CONTENTS