MH1201 Linear Algebra II - AY 2014-2015 (S2) - Lecture Notes Summary - MH1201 LINEAR ALGEBRA II AY 2014-2015 Dr Le Hai Khoi([email protected] Division

# MH1201 Linear Algebra II - AY 2014-2015 (S2) - Lecture Notes Summary

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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. 3
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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 1
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Chapter 1 Vector Spaces In MH1200, we already know how the set R n of all n -tuples of real numbers, together with the coordinate-wise addition and scalar multiplication operations defined on it, has the same algebraic properties as the familiar algebra of geometric vectors. The goal of this chapter is to develop a notion of vectors that encompasses not only physical interpretations (forces, velocities, accelerations, ...) and geometric interpretations (vectors), but also the many mathematical applications. 1.1 Basic Notation Let V be a non-empty set, whose elements are called vectors , and R the set of real numbers, whose elements are called scalars . Let further, on V two operations (called addition and scalar multiplication ) are defined in the following sense: - Vectors can be added: we use the usual sign + to denote an addition operation, and the result of adding two vectors u, v V is denoted by u + v . - Vectors can be multiplied by scalars: we use the usual notation λu to denote the result of scalar multiplying the vector u V by the scalar λ R . Note that the vector addition and/or the scalar multiplication, in general, will not yield another vector in V . For example, if V = Z , the set of all integers, then the usual operations of addition and scalar multiplication define addition and scalar multiplication operations on V . However, λu is not always in V .