Linear Algebra, Theory And Applications

1 Linear Algebra, Theory And ApplicationsKenneth KuttlerJanuary 29, 2012 Saylor URL: Saylor Foundation2 Saylor URL: Saylor FoundationContents1 Sets And Set Notation .. Functions .. The Number Line And Algebra Of The Real Numbers .. Ordered fields .. The Complex Numbers .. Exercises .. Completeness ofR.. Well Ordering And Archimedean Property .. Division And Numbers .. Systems Of Equations .. Exercises .. Algebra inFn.. Exercises .. The Inner Product InFn.. What Is Linear Algebra? .. Exercises .. 362 Matrices And Linear Matrices .. TheijthEntry Of A Product .. Digraphs .. Properties Of Matrix Multiplication .. Finding The Inverse Of A Matrix .. Exercises .. Linear Transformations .. Subspaces And Spans .. An Application To Matrices .. Matrices And Calculus .. The Coriolis Acceleration.

2 The Coriolis Acceleration On The Rotating Earth .. Exercises .. 713 Basic Techniques And Properties .. Exercises .. The Mathematical Theory Of Determinants .. The Function sgn .. 843 Saylor URL: Saylor TheDefinition Of The Determinant .. A Symmetric Definition .. Basic Properties Of The Determinant .. Expansion Using Cofactors .. A Formula For The Inverse .. Rank Of A Matrix .. Summary Of Determinants .. The Cayley Hamilton Theorem .. Block Multiplication Of Matrices .. Exercises .. 1024 Row Elementary Matrices .. The Rank Of A Matrix .. The Row Reduced Echelon Form .. Rank And Existence Of Solutions To Linear Systems .. Fredholm Alternative .. Exercises .. 1185 some .. Finding AnLUFactorization .. Solving Linear Systems Using AnLUFactorization .. ThePLUF actorization.

3 Justification For The Multiplier Method .. Existence For ThePLUF actorization .. TheQRFactorization .. Exercises .. 1336 Linear Simple Geometric Considerations .. The Simplex Tableau .. The Simplex Algorithm .. Maximums .. Minimums .. Finding A Basic Feasible Solution .. Duality .. Exercises .. 1567 Spectral Eigenvalues And Eigenvectors Of A Matrix .. some Applications Of Eigenvalues And Eigenvectors .. Exercises .. Schur s Theorem .. Trace And Determinant .. Quadratic Forms .. Second Derivative Test .. The Estimation Of Eigenvalues .. Advanced Theorems .. Exercises .. 190 Saylor URL: Saylor FoundationCONTENTS58 Vector Spaces And Vector Space Axioms .. Subspaces And Bases .. Basic Definitions .. A Fundamental Theorem .. The Basis Of A Subspace .. Lots Of Fields .. Irreducible Polynomials.

4 Polynomials And Fields .. The Algebraic Numbers .. The Lindemannn Weierstrass Theorem And Vector Spaces .. Exercises .. 2199 Linear Matrix Multiplication As A Linear Transformation .. (V,W) As A Vector Space .. The Matrix Of A Linear Transformation .. some Geometrically Defined Linear Transformations .. Rotations About A Given Vector .. The Euler Angles .. Eigenvalues And Eigenvectors Of Linear Transformations .. Exercises .. 24210 Linear Transformations Canonical A Theorem Of Sylvester, Direct Sums .. Direct Sums, Block Diagonal Matrices .. Cyclic Sets .. Nilpotent Transformations .. The Jordan Canonical Form .. Exercises .. The Rational Canonical Form .. Uniqueness .. Exercises .. 27311 Markov Chains And Migration Regular Markov Matrices .. Migration Matrices .. Markov Chains .. Exercises .. 28412 Inner Product General Theory .

5 The Gram Schmidt Process .. Riesz Representation Theorem .. The Tensor Product Of Two Vectors .. Least Squares .. Fredholm Alternative Again .. Exercises .. The Determinant And Volume .. Exercises .. 306 Saylor URL: Saylor Foundation6 CONTENTS13 SelfAdjoint Simultaneous Diagonalization .. Schur s Theorem .. Spectral Theory Of Self Adjoint Operators .. Positive And Negative Linear Transformations .. Fractional Powers .. Polar Decompositions .. An Application To Statistics .. The Singular Value Decomposition .. Approximation In The Frobenius Norm .. Squares And Singular Value Decomposition .. Moore Penrose Inverse .. 33414 Norms For Finite Dimensional Vector ThepNorms .. The Condition Number .. The Spectral Radius .. Series And Sequences Of Linear Operators .. Iterative Methods For Linear Systems .. Theory Of Convergence.

6 Exercises .. 36315 Numerical Methods For Finding The Power Method For Eigenvalues .. The Shifted Inverse Power Method .. The Explicit Description Of The Method .. Complex Eigenvalues .. Rayleigh Quotients And Estimates for Eigenvalues .. TheQRAlgorithm .. Basic Properties And Definition .. The Case Of Real Eigenvalues .. TheQRAlgorithm In The General Case .. Exercises .. 401A Positive Matrices403B Functions Of Matrices411C Applications To Differential Theory Of Ordinary Differential Equations .. Linear Systems .. Local Solutions .. First Order Linear Systems .. Geometric Theory Of Autonomous Systems .. General Geometric Theory .. The Stable Manifold .. 434D Compactness And The Nested Interval Lemma .. Convergent Sequences, Sequential Compactness .. 440 Saylor URL: Saylor FoundationCONTENTS7E TheFundamental Theorem Of Algebra443F Fields And Field The Symmetric Polynomial Theorem.

7 The Fundamental Theorem Of Algebra .. Transcendental Numbers .. More On Algebraic Field Extensions .. The Galois Group .. Normal Subgroups .. Normal Extensions And Normal Subgroups .. Conditions For Separability .. Permutations .. Solvable Groups .. Solvability By Radicals .. 482G Answers To Selected Exercises .. Exercises .. Exercises .. Exercises .. Exercises .. Exercises .. Exercises .. Exercises .. Exercises .. Exercises .. Exercises .. Exercises .. Exercises .. Exercises .. Exercises .. Exercises .. Exercises .. Exercises .. Exercises .. Exercises .. Exercises .. Exercises .. Exercises .. 496 Copyrightc 2012,Saylor URL: Saylor Foundation8 CONTENTSS aylor URL: Saylor FoundationPrefaceThis isa book on Linear algebra and matrix Theory . While it is self contained, it will workbest for those who have already had some exposure to Linear algebra.

8 It is also assumed thatthe reader has had calculus. some optional topics require more analysis than this, think that the subject of Linear algebra is likely the most significant topic discussed inundergraduate mathematics courses. Part of the reason for this is its usefulness in unifyingso many different topics. Linear algebra is essential in analysis, applied math, and even intheoretical mathematics. This is the point of view of this book, more than a presentationof Linear algebra for its own sake. This is why there are numerous Applications , some book features an ugly, elementary, and complete treatment of determinants earlyin the book. Thus it might be considered as Linear algebra done wrong. I have done thisbecause of the usefulness of determinants. However, all major topics are also presented inan alternative manner which is independent of book has an introduction to various numerical methods used in Linear is done because of the interesting nature of these methods.

9 The presentation hereemphasizes the reasons why they work. It does not discuss many important numericalconsiderations necessary to use the methods effectively. These considerations are found innumerical analysis the exercises, you may occasionally see at the beginning. This means you ought tohave a look at the exercise above it. some exercises develop a topic sequentially. There arealso a few exercises which appear more than once in the book. I have done this deliberatelybecause I think that these illustrate exceptionally important topics and because some peopledon t read the whole book from start to finish but instead jump in to the middle is one on a theorem of Sylvester which appears no fewer than 3 times. Then it is alsoproved in the text. There are multiple proofs of the Cayley Hamilton theorem, some in theexercises. some exercises also are included for the sake of emphasizing something which hasbeen done in the preceding URL: Saylor Foundation10 CONTENTSS aylor URL: Saylor SetsAnd Set NotationA set is just a collection of things called elements.

10 For example{1,2,3,8}would be a setconsisting of the elements 1,2,3, and 8. To indicate that 3 is an element of{1,2,3,8},it iscustomary to write 3 {1,2,3,8}.9/ {1,2,3,8}means 9 is not an element of{1,2,3,8}.Sometimes a rule specifies a set. For example you could specify a set as all integers largerthan would be written asS={x Z:x >2}.This notation says: the set of allintegers,x,such thatx > sets with the property that every element ofAis an element ofB,thenAisa subset example,{1,2,3,8}is a subset of{1,2,3,4,5,8},in symbols,{1,2,3,8} {1,2,3,4,5,8}.It is sometimes said that Ais contained inB or even BcontainsA .The same statement about the two sets may also be written as{1,2,3,4,5,8} {1,2,3,8}.The union of two sets is the set consisting of everything which is an element of at leastone of the sets,AorB. As an example of the union of two sets{1,2,3,8} {3,4,7,8}={1,2,3,4,7,8}because these numbers are those which are in at least one of the two sets.