Transcription of Linear Algebra I - Lectures Notes - Spring 2013
1 Linear Algebra I - Lectures Notes - Spring 2013 Shmuel FriedlandOffice: 715 SEO, phone: 413-2176,e:mail: friedlanApril 28, 2013 Remarks on the Notes : These Notes are updated during the course. For2012 version view : Linear Algebra is used in many areas as: engineering, biology,medicine, business, statistics, physics, mathematics, numerical analysis andhumanities. Thereason: many real world systems consist of many partswhich interact linearly. Analysis of such systems involves the notions and thetools from Linear Algebra .
2 These Notes of Linear Algebra course emphasize themathematical rigour over the applications, contrary to many books on linearalgebra for engineers. My main goal in writing these Notes was to give to thestudent a concise overview of the main concepts,ideas and results that usuallyare covered in the first course on Linear Algebra for mathematicians. Thesenotes should be viewed as a supplementary Notes to a regular book for linearalgebra, as for example [1].Main Topics of the Course SYSTEMS OF EQUATIONS VECTOR SPACES Linear TRANSFORMATIONS DETERMINANTS INNER PRODUCT SPACES EIGENVALUES JORDAN CANONICAL FORM-RUDIMENTSText: Jim Hefferon, Linear Algebra , andSolutionsAvailable for free : MatLab,Maple, Matematica1 Contents1 Linear equations and System of Linear Equation.
3 Systems of equations .. operations .. systems .. Matrix formalism for solving systems of Linear equations .. Coefficient Matrix of the system .. row operations .. Echelon Form .. of Linear systems .. row echelon form .. Operations on vectors and matrices .. on vectors .. to solutions of Linear systems .. of matrices with vectors .. equivalence of matrices ..182 Vector Definition of a vector space .. Examples of vector spaces .. Subspaces .. Examples of subspaces .. Linear combination & span.
4 Spanning sets of vector spaces .. Linear Independence .. Basis and dimension .. Row and column spaces of matrices .. An example for a basis ofN(A) .. Useful facts .. The spacePn.. Sum of two subspaces .. Sums of many subspaces .. Fields .. Finite Fields .. Vector spaces over fields ..313 Linear One-to-one and onto maps .. Isomorphism of vector spaces .. Iso. of fin. dim. vector spaces .. Isomorphisms ofRn.. Examples .. Linear Transformations (Homomorphisms) .. Rank-nullity theorem.
5 Matrix representations of Linear transformations .. Composition of maps .. Product of matrices .. Transpose of a matrixA>.. Symmetric Matrices .. Powers of square matrices and Markov chains .. Inverses of square matrices .. Elementary Matrices .. ERO in terms of Elementary Matrices .. Matrix inverse as products of elementary matrices .. Gauss-Jordan algorithm forA 1.. Change of basis .. An example .. Change of the representation matrix under the change of Example .. Equivalence of matrices ..484 Inner product Scalar Product inRn.
6 Cauchy-Schwarz inequality .. Scalar and vector projection .. Orthogonal subspaces .. Example .. Projection on a subspace .. Example .. Finding the projection on span .. The best fit line .. Example .. Orthonormal sets .. Orthogonal Matrices .. Gram-Schmidt orthogonolization process .. Explanation of G-S process .. An example of G-S process .. QR Factorization .. An example of QR algorithm .. Inner Product Spaces .. Examples of IPS .. Length and angle in IPS .. Orthonormal sets in IPS.
7 Fourier series .. Short biographies of related mathematcians .. Johann Carl Friedrich Gauss .. Augustin Louis Cauchy .. Hermann Amandus Schwarz .. Viktor Yakovlevich Bunyakovsky .. Gram and Schmidt .. Jean Baptiste Joseph Fourier .. J. Peter Gustav Lejeune Dirichlet .. David Hilbert ..645 Introduction to determinant .. Determinant as a multilinear function .. Computing determinants using elementary row or columns op-erations .. Permutations .. S2.. S3.. Rigorous definition of determinant.
8 Casesn= 2,3 .. Minors and Cofactors .. Examples of Laplace expansions .. Adjoint Matrix .. The properties of the adjoint matrix .. Cramer s Rule .. History of determinants ..766 Eigenvalues and Definition of eigenvalues and eigenvectors .. Examples of eigenvalues and eigenvectors .. Similarity .. Characteristic polynomials of upper triangular matrices .. Defective matrices .. An examples of a diagonable matrix .. Powers of diagonable matrices .. Systems of Linear ordinary differential equations.
9 Initial conditions .. Complex eigenvalues of real matrices .. Second Order Linear Differential Systems .. Exponential of a Matrix .. Examples of exponential of matrices ..871 Linear equations and matricesThe object of this section is to study a set ofmlinear equations innrealvariablesx1,..,xn. It is convenient to groupnvariable in one quantity:(x1,x2,..,xn), which is called arow vector. For reasons that will be seen4later we will consider acolumnvector denoted byx, for which we either theround brackets ( ) or the straight brackets [ ]:x:= (x1,x2.)
10 ,xn)>= = .( )We will denote byx>the row vector (x1,x2,..,xn). We denote the set ofall column vectorsxbyRn. SoR1=Rall the points on the real line;R2are all points in the plane;R3all points in 3-dimensional calledn-dimensionalspace. It is hard to visualizeRnforn 4, but we ca study itefficiently using mathematical will learn how to determine when a given system is Linear equations inRnis unsolvable or solvable, when the solution is unique or not unique, andhow to express compactly all the solutions of the given System of Linear Equationa11x1+a12x2+.
