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# LINEAR ALGEBRA MATH 355

Rice University

GPA 3.86

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This 5 page Class Notes was uploaded by Jayde Lang on Monday October 19, 2015. The Class Notes belongs to MATH 355 at Rice University taught by Staff in Fall. Since its upload, it has received 55 views. For similar materials see /class/224939/math-355-rice-university in Mathematics (M) at Rice University.

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Date Created: 10/19/15

Math 355 Final Review Sheet Solutions of systems of linear equations What techniques do we have to nd these solutions when they exist existence When are they unique uniqueness Homogeneous vs nonhomogeneous Parametric vector form of a solution Different representations of systems of linear equations Vector equation Matrix equation Matrices Properties of matrix addition scalar multiplication Matrix multiplication different representations e g rowcolumn multiplication vs writing columns Properties of matrix multiplication Transpose of a matrix properties Inverse of a matrix How can you tell if a matrix is invertible The Invertible Matrix Theorem Determinant of a matrix cofactor expansion properties Cramer s Rule Vector spaces De nition Examples Dimension of a vector space Subspaces De nition of a subspace Spanv1 vquot linear combinations Column space Null space differences between these p 232 Rank of a matrix The Rank Theorem Linear dependence and linear independence De nitions Useful theorems Basis The Spanning Set Theorem The Basis Theorem Linear transformations De nitions Domain range codomain kernel Matrix transformation When is a transformation linear Matrix of a linear transformation standard matrix Onetoone and onto How did we prove these Matrix for a linear transformation relative to bases 8 and C Coordinate Systems Coordinate vector of x relative to a basis 8 The Unique Representation Theorem Change of coordinates matrix change of basis Coordinate mapping Eigenvalues and eigenvectors De nitions Eigenspace Characteristic polynomial characteristic equation multiplicity of an eigenvalue Finding eigenvalues and eigenvectors Complex eigenvalues Diagonalization Similar matrices The Diagonalization Theorem Steps to diagonalize a matrix if possible Diagonal Matrix Representation Theorem Diagonalization of Symmetric Matrices Orthogonally diagonalizable The Spectral Theorem Inner product spaces Dot product in Rquot properties General inner product properties Lengthnorm unit vectors Normalizing Distance between vectors Orthogonal vectors orthogonal sets amp bases orthonormal sets amp bases Orthogonal complement of a subspace W Orthonormal columns of a matrix Properties of orthogonal sets Theorem 5 p 385 Orthogonal projection Orthogonal Decomposition Theorem Best Approximation Theorem GramSchmidt process CauchySchwarz Inequality Triangle Inequality Quadratic Forms De nition matrix of a quadratic form Change of variable The Principal Axes Theorem Positive de nite negative de nite inde nite Jordan Canonical Form Almost diagonal what does the matrix look like For the right basis as the columns of P A PJP39 Generalized eigenvectors generalized eigenspaces Cycle of generalized eigenvectors Similar matrices have the same Jordan canonical form Proof techniques we ve talked about For an IfThen statement assume the If prove the Then Proof by contrapositive Proof by contradicton Proof by induction Proof of if and only if statements use a circle or a bunch of little circles Other notes about proofs Always use all of your assumptions If you re stuck start with the definitions often times the proofs in the book are the slickest way to do things but maybe not the only way Make sure you are explicit about the logic and reasoning in your proofs MATH 355 Linear Algebra7 Summer 2005 Review Sheet 1 Vector Spaces 7 vector space subspace linear combination span linear dependence basis dimension 7 Subsets of linearly independent sets are linearly independent 7 Replacement Theorem 7 If V is n dimensional any linearly independent set containing n elements is a basis 7 The dimension of a subspace is always less that or equal to the dimension of the space 2 Linear Transformations and Matrices 7 linear transformation null space range rank nullity ordered basis coordinate vector matrix representations of linear transformations left multiplication transformation invertibility inverse change of coordinates matrix 7 T V 7gt W linear v1luvn a basis for V then RT spanTv1 l Tvnl 7 T V 7gt W linear dimV nullityT rankTl 7 T V 7gt W linear dimV dimW T is onetoone if and only if T is onto 7 Properties of the matrix representations with respect to ordered bases 7 T V 7gt W linearl If T is invertible then dimV dimWl 7 For nite dimensional vector spaces V W over the same base eld V and W are isomorphic if and only if dimV dimW 3 Elementary Matrix Operations and Systems of Linear Equations 7 three types of elementary operations elementary matrices rank of a matrix inverse of a matrix Gaussian Elimination reduced row echelon form 7 Elementary row and column operations are rank preservingl 7 The rank of a matrix equals the number of linearly independent columns rows 7 properties of rank 7 Every invertible matrix is the product of elementary matricesl 7 Let Az 0 be a system of m linear equations and n unknowns If m lt n the system has a nonzero solution 7 Let Az b be a system of n linear equations and n unknowns A is invertible if and only if the system has exactly one solution 7 The system Az b has at least one solution if and only if rankA rankA l b 4 Determinants de nition of an n X n determinant det AB det A det Let E be en elementary matrix 1 if E is of type 1 detE k if E is of type 2 with one row of I multiplied by k 1 if E is of type 3 7 detA detA 7 A is invertible if and only if detA 0i 7 The determinant of an upper triangular matrix is the product of the diagonal entries 7 If A is invertible then detA 1 detlwy MATH 355 Linear Algebra7 Summer 2005 Review Sheet 5 Diagonalization 7 diagonalizable similar matrices eigeh alne 39 t 39 1 split 1 39 character istic polynomial multiplicity invariant subspaces 7 T E V is diagonalizable if and only if there exists an ordered basis for V consisting of eigen vectors 0 i 7 The characteristic polynomial of an n X n matrix is a polynomial of degree n With leading coef cient 71 i 7 Test for the diagonalizability of T E V 1 The characteristic polynomial splits over the base eld 2 For each eigenvalue A the multiplicity of A equals dimE n 7 rankT 7 Al 7 CayleyHamilton Theorem 6 Inner Product Spaces 7 inner product adjoint of a matrix norm inner product space orthogonal unit vector orthonor mal orthonormal basis orthogonal complement orthogonal projection adjoint of a transfor mation normal transformation and matrix selfadjoint unitary operator matrix orthogonal operatormatrix unitarily orthogonally equivalent 7 GramSchmidt Orthogonalization Process 7 A set S 111112 i i is orthonormal if and only if v1 vj 617x 7 An orthogonal set of nonzero vectors is linearly independent 7 If W E V then dimV dimW dimWil 7 Schur s Theorem 7 Equivalences to being unitary or orthogonali 7 Every eigenvalue of a selfadjoint operator is real 7 Let T E V Where V is a nite dimensional real inner product space Then T is both self adjoint and symmetric if and only if there exists an orthonormal basis for V of eigenvectors of T corresponding to eigenvalue of absolute value 1 7 Let T E V Where V is a nite dimensional complex inner product space Then T is unitary if and only if there exists an orthonormal basis for V of eigenvectors of T corresponding to eigenvalue of absolute value 1 7 Let A E MnxnCl A is normal if and only if A is unitarily equivalent to a diagonal matrix 7 Let A E MnxnRl A is symmetric if and only if A is orthogonally equivalent to a diagonal matrix 7 S CauchySchwartz lnequality 7 Mac W HIM M Triangle Inequality 7 Jordan Canonical Form 7 Jordan block Jordan canonical form Jordan canonical basis generalized eigenvectors and eigenspaces cycle of generalized eigenvectors dot diagrams 7 For a xed eigenvalue A of T E V dimK equals the multiplicity of Al 7 The number of Jordan blocks corresponding to an eigenvalue A equals dimEl 7 Let T be a linear transformation on a nite dimensional vector such that the characteristic poly nomial of T splitsl Then T is diagonalizable if and only if K T EA for each eigenvalue A of Ti 7 If A is an n X n matrix Whose characteristic polynomial splits then A has a Jordan canonical form J and A is similar to i 7 Two matricies are similar if and only if they have the same Jordan canonical form up to the ordering of the eigenvalues

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