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1 Chapter Ax = b system is solved by Augmented Matrix Solutions from Ax = 0 is like this: [ x ] = [ p ] + x3 [ v ] Theorem 4 All of these are all true and all false at the same time 1) Ax = b has solution for each b 2) b is linear combinations of col A 3) columns of A span Rm 4) pivot position in every row of row reduced matrix Theorem 5 Conditions for Linear Operator 1) A (u + v) = Au + Bv 2) A (cu) = c Au Theorem 6 v is solution to Ax = 0 p is solution to Ax = b Theorem 8 If a set has more vectors than entries in each vector, then the set is linearly dependent Theorem 10 Uniqueness Matrix Each T (x) has a unique A such that T (x) = Ax A is in form A = [ T (e1) T (e2) T (e3) ... T (en)] Matrix Transformations In rotation, [1] = [cos u] A = [ cos u - sin u ] [0] [sin u] [ sin u cos u ] Onto Transformation If each b in Rm is image of at least one x in Rn One to One Transformation If each b in Rm is image of at most one x in Rn Theorem 11 T is one to one iff T (x) = 0 has only trivial solution Theorem 12 A) T maps Rm onto Rn iff columns of A span Rn B) T is one to one iff col A is linearly independent Chapter 2 Theorem 3 (A + B)T = A T + BT (AB)T = BTA T x = A-1b is solution Theorem 6 (A-1) -1 = A (AB) -1 = B-1A-1 (A T) -1 = (A-1) T Elementary Matrix A single elementary row operation on I Elementary row operation performed on A is EA, where E is made by doing same operation on I E is invertible, inverse of E is elementary matrix that makes E back to I Theorem 7 Matrix A is invertible iff A is row equivalent to I, and elementary row operations that reduces A to I also makes I into A -1 Invertible Matrix Theorem All of these are all true and all false at the same time A) row equivalence to I B) n pivots C) Ax = 0 has only trivial solution D) linearly independent columns F) Ax = b has at least one solution for each b G) columns span Rn H) Ax is one to one J) there exists D such that AD = I K) A T is invertible L) columns form basis of Rn M) Col A = Rn O) Nul A = {0} P) dim Nul A = 0 Q) det A != 0 R) number 0 is not eigenvalue of A Homogenous Coordinates Action Operator with 1 added to bottom left and vectors has 1 tagged on Translation vector occurs on left column Subspace Contains {0} vector and is close under both multiplication by scalars and addition by vectors Nullspace Set Nul A of all solutions to Ax = 0 Theorem 12 Nul A is subspace of A Theorem 13 Pivot column forms basis for Colspace [x]B Coordinate Vector Dimension Number of vectors in basis Rank Dimension of Colspace Theorem 14 rank A + dim Nul A = number of columns Theorem 15 Basis Theorem Any linearly independent set of p elements in H where dim H = p is basis for H E) Ax maps Rn onto Rn I) there exists C such that CA = I N) dim Col A = rank = n Chapter 3 Theorem 1 det A = ai1C i1 + ai2Ci2 + ... + ainCin where Cij = (-1)^(i+j) det Aij where aij are elements of matrix A Theorem 3 Row Operations and Determinants A) multiple of one row is added to another row to make B then det B = det A B) two rows switched to make B then det B = det A C) one row multiplied by k to make B then det B = k det A T Theorem 5 det A = det A Theorem 6 det (AB) = (det A) (det B) Theorem 7 Cramer's Rule In solution vector x, xi = (det Ai(b)) / (det A) where b substitutes for col i Classical Adjoint Matrix of Cofactors Theorem 8 A-1 = (1/det A) adj A Theorem 9 Areas and Volumes A) Area of parallelogram determined by columns of A is | det A | B) Volume of parallelepiped determined by columns of A is | det A | Theorem 10 Areas and Volumes Under Linear Transformation A) {area of T (S)} = | det A | {area of S} B) {volume of T (s) = | det A | {volume of S} Chapter 4 Vector Space A) u + v is in V B) commutative law holds C) associative law holds D) zero vector exists F) closed under multiplication by scalars G) operator is distributive H) vector is distributive H) c(du) = (cd)u Theorem 1 If v1, v2, ... vp are in vector space V, then their span is a subspace of V Kernel Nullspace Range set of all vectors in W of the form T (x) for some x in V Kernel and Range If T happens as matrix transformation, then kernel and range of T are respectively nullspace and colspace Theorem 4 Set of 2 and more vectors are lin dependent iff one vector is a linear combination of the other vectors Basis A set of lin independent vectors that coincides with A by the subspace it spans Theorem 5 Spanning Set Theorem A) set still spans if a lin dependent vector is removed from set B) some subset of a vector that spans V is a basis for V Elementary Row Operations Elementary row operations do not affect linear dependence relations among the columns of the matrix Theorem 7 If B is a basis for vector space V, for each x in V, there exists a unique linear combination of vectors in B that would add up to x Change of Basis x = PB [x] B where PB is the change-of-coordinates matrix Theorem 8 Coordinate mapping of x into B if B is a basis for vector space V is a one-to-one linear transformation from V onto Rn. Isomorphism Every vector space calculation in V is accurately reproduced in W and vice versa. Theorem 11 Subspace has lower dimension than original vector space Theorem 12 Basis Theorem Any linearly independent set of p vectors where dim V = p is automatically a basis. Any set of exactly p elements that spans V is automatically a basis. dim of Nul A is number of free variables and dim of Col A is number of pivot columns in A Theorem 18 Markov Chains If P is a regular stochastic matrix, P then has a unique steady-state vector q. E) existence of u for every u I) 1u = u Chapter 5 Theorem 1 Eigenvalues of triangular matrix are entries on its main diagonal. Theorem 2 Eigenvectors correspond to distinct eigenvalues are linearly independent. det A = (-1)^r (products of pivots in U, row reduced A) when invertible = 0 when not invertible Theorem 4 Similarity (B = P A P-1) 2 similar matrices have same characteristic polynomial and same eigenvectors Theorem 5 Diagonalization Theorem A) n by n matrix A is diagonalizable iff A has n linearly independent eigenvectors B) A = P D P-1 where P are eigenvectors and D are corresponding eigenvalues Theorem 6 n by n matrix with n distinct eigenvalues is diagonalizable Theorem 7 A) Dimension of the eigenspace for a particular eigenvalue is less than or equal to the eigenvalue's multiplicity B) Matrix A is diagonalizable iff sum of dimensions of distinct eigenspaces equals n (i.e. multiplicity of each eigenvalue = dimension of its eigenspace) C) All the bases of each eigenspace of each eigenvalue combine to form an eigenvector basis for R n Theorem 8 Diagonal Matrix Representation If A = P D P-1 and if B is the basis for Rn formed from the columns of P, then D is the matrix for the transformation x into Ax Complex vectors have form x = Re x + Im x and the complex conjugate has xbar = Re x Im x Axbar = (Ax)bar = ( x)bar = barxbar shows that if A is real complex eigenvalues occur in conjugate pairs Theorem 9 If 2 by 2 matrix A has complex eigenvalue a bi, then A = P D P-1 where P = [ Re v Im v ] and C = [ a -b ] [b a] Dynamical Systems A) xk+1 = Axk B) A is the coefficient matrix C) xk is in form c1( 1)kv1 + .... + cn( n)kvn Differential Equations A) x' = Ax B) x is in form v1e 1t + .... + vne nt C) Ax is in form Av1e 1t + .... + Avne nt D) x' is in form Differential Equations with Complex Eigenvalues at at A) y1(t) = Re x1(t) = [(Re v) cos bt (Im v) sin bt] e B) y2(t) = Im x1(t) = [(Re v) sin bt + (Im v) cos bt] e Decoupling Making A an diagonal matrix Let y = P-1 x then d/dt (P y) = A (P y) = PDy x(t) therefore is in form c1v1e 1t + .... + cnvne nt Power Method A) Choose x0 whose largest entry is 1 B) For k = 0, 1, .... a. calculate Axk b. Let k be entry in Axk whose absolute value is as large as possible c. Compute xk+1 = (1/ k) Axk C) k approaches dominant eigenvalue while xk approaches corresponding eigenvector Inverse Power Method Select an estimate close enough to the eigenvalue and select an initial eigenvector whose largest entry is 1. Solve the system (A I) yk xk. Let xk+1 = (1/ k) yk Chapter 6 Theorem 1 A) u v = v u Magnitude of Vector Distance of Vectors Theorem 3 Theorem 4 Theorem 5 Projection Theorem 6 Theorem 7 A) ||Ux|| = ||x|| Theorem 8 Best Approximation Theorem Theorem 10 Gram-Schmidt Algorithm 1v1e 1t + .... + nvne nt yk = P-1 xk and yk+1 = D yk k be entry in yk whose absolute value is as large as possible. Compute k = + (1/ k) and B) (u + w) v = u v + w v C) c(u) v = c (u v) = u (cv) D) u u 0 ||u|| = (u u)^(1/2) dist (u, v) = ||u v|| (Row A) = Nul A (Col A) = Nul A T If S is an orthogonal set, then S is linearly independent set and is a basis for the subspace spanned by S y = c1v1 + c2v2 + .... + cnvn cn = (y un)/(un un) yhat = proju y = (y y)/(u u) u T U U = I when U has orthonormal columns when U is orthonormal B) (Ux) (Uy) = x y C) (Ux) (Uy) = 0 iff x y = 0 y = yhat + z where z is in W yhat is the best approximate of y such that ||y-yhat|| < ||y v|| for any v in W If U is orthonormal set, then projwy = UUT y v1 = x1 v2 = x2 (x2 v1)/(v1 v1) v1 v3 = x3 (x3 v1)/(v1 v1) v1 (x3 v2)/(v2 v2) v2 Such that Span of vector set v = Span of vector set x Theorem 12 QR Factorization A = QR where Q have orthonormal basis for Col A and R is upper triangular matrix where Q TA = Q T(QR) = IR = R Least Squares Solutions xhat is solution of ATAx = A Tb Theorem 14 In matrix A TA is invertible iff columns of A are lin independent, which means that Ax = b has only one least squares solution given by xhat = (A TA) -1ATb Theorem 15 If A = QR, then xhat = (R) 1QTb Least Squares Line Let X = [1 x coordinates] and y = y coordinates. is solution of XTX = XTy. The individual components of will form the coefficients of the polynomial. (Note also: y = X Inner Product Space A) <u, v> = <v, u> B) <u + w, v> = <u, v> + <w, v> C) <cu, v> = c <u, v> = D) <u, u> 0 unless u = 0 E) <p, q> = INT (from a to b) p (t) q (t) dt Cauchy-Schwarz Inequality |<u, v>| ||u|| ||v|| Triangle Inequality ||u + v|| ||u|| + ||v|| Fourier Series ak = <f, cos kt>/<cos kt, cos kt> bk = <f, sin kt>/<sin kt, sin kt> Chapter 7 Theorem 1 If A is symmetric, then any 2 eigenvectors from different eigenspaces are orthogonal. Theorem 2 An n by n matrix A is orthogonally diagonalizable iff A is symmetric matrix. Theorem 3 Spectral Theorem An n by n matrix has the following properties A has n real eigenvalues, counting multiplicities. The dimension of the eigenspace for each eigenvalue equals the multiplicity of as a root of the characteristic equation. The eigenspaces are mutually orthogonal, in th sense that eigenvectors corresponding to different eigenvalues are orthogonal. A is orthogonally diagonalizable. Spectral Decomposition A = 1u1u1 + .... + nunun where u are (unit and orthogonal) eigenvectors of A Projection Matrix (unun ) is orthogonal projection matrix projecting on space
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Path: Cornell >> COM S >> 280 Fall, 2005
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Path: Cornell >> BEE >> 102 Fall, 2005
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Path: Cornell >> BEE >> 102 Fall, 2005
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Path: Cornell >> CHEM >> 2080 Spring, 2005
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Path: Cornell >> CHEM >> 2080 Spring, 2005
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Path: Cornell >> CHEM >> 2080 Spring, 2005
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Path: Cornell >> CHEM >> 2080 Spring, 2005
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Path: Cornell >> NBA >> 6890 Fall, 2007
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Path: Cornell >> PSYCH >> 1101 Fall, 2006
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Path: Cornell >> PSYCH >> 1101 Fall, 2006
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Path: Cornell >> PSYCH >> 1101 Fall, 2006
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Path: Cornell >> H ADM >> 115 Spring, 2006
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Path: Cornell >> H ADM >> 115 Spring, 2006
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Path: Cornell >> H ADM >> 115 Spring, 2006
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Path: Cornell >> AM ST >> 202 Spring, 2007
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Path: Cornell >> H ADM >> 174 Fall, 2005
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Path: Cornell >> H ADM >> 174 Fall, 2005
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Path: Cornell >> D SOC >> 101 Fall, 2006
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Path: Cornell >> D SOC >> 101 Fall, 2006
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Path: Cornell >> D SOC >> 101 Fall, 2006
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Path: Cornell >> D SOC >> 101 Fall, 2006
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Path: Cornell >> D SOC >> 101 Fall, 2006
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Path: Cornell >> D SOC >> 101 Fall, 2006
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Path: Cornell >> D SOC >> 101 Fall, 2006
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Path: Cornell >> D SOC >> 101 Fall, 2006
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Path: Cornell >> ECON >> 1110 Fall, 2006
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Path: Cornell >> ECON >> 1110 Fall, 2006
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Path: Cornell >> ECON >> 1110 Fall, 2006
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Path: Cornell >> ECON >> 1110 Fall, 2006
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Path: Cornell >> ECON >> 1110 Fall, 2006
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Path: Cornell >> ECON >> 1110 Fall, 2006
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Path: Cornell >> ECON >> 1110 Fall, 2006
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Path: Cornell >> ECON >> 1110 Fall, 2006
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Path: Cornell >> ECON >> 1110 Fall, 2006
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Path: Cornell >> PAM >> 2300 Fall, 2006
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Path: Cornell >> PAM >> 2300 Fall, 2006
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Path: Cornell >> PAM >> 2300 Fall, 2006
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Path: Cornell >> PAM >> 2300 Fall, 2006
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Path: Cornell >> PAM >> 2300 Fall, 2006
Description: Background: Attorney General Janet Reno said in 2000 that prisoner reentry is one of the most pressing problems we face as a nation. Since the year 2000, however, the problem has not improved, a national study showed that in 2005, 3 years after 600,0...
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Path: Cornell >> PAM >> 2300 Fall, 2006
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Path: Cornell >> GOVT >> 3303 Spring, 2006
Description: Balloga, 1 Abram Balloga Lowell Turner Govt 330 6/13/2007 This Social Europe The concept of social Europe arose as a rebuilding and \"never again\" measure after World War II. The European Union was created with the purpose of uniting European countrie...
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Path: Cornell >> GOVT >> 3303 Spring, 2006
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Path: Cornell >> AEM >> 2200 Spring, 2007
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Path: Cornell >> AEM >> 2200 Spring, 2007
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Path: Cornell >> CHEM >> 1160 Spring, 2007
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Path: Cornell >> CHEM >> 1160 Spring, 2007
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Path: Cornell >> CHEM >> 1160 Spring, 2007
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aem120-9-19
Path: Cornell >> AEM >> 1200 Fall, 2006
Description: 9/19 Chapter 4,9: Doing Market Research in the Entrepreneurial Startup The need for market information Gathering information o Primary data o Secondary data Paul Joseph (Simon Management Group) Why failure? o Many entrepreneurs who found new venture...
aem120-8-24
Path: Cornell >> AEM >> 1200 Fall, 2006
Description: 8/24 Chapter 1 What is Entrepreneurship? What is an entrepreneur? o One who organizes, manages, and assumes the risks of a business or enterprise o Business entrepreneurs are often highly regarded in US culture as being a critical component of its ...
aem120-8-29
Path: Cornell >> AEM >> 1200 Fall, 2006
Description: 8/29 Chapter 2: Change and Entrepreneurship Change as a catalyst of new entrepreneurship o New technology Info technology/biotech o Political/regulatory shifts globalization o Social/demographic change Development of a \"mass market\" into a \"consu...
aem120-8-31
Path: Cornell >> AEM >> 1200 Fall, 2006
Description: 8/31 Chapter 2: Business Economics of Entrepreneurship Factors influencing the success of new firms New v. Established Firms Competency destroying change Entrepreneurs who create new businesses primarily focus on o New products, services, and market...
aem120-9-5
Path: Cornell >> AEM >> 1200 Fall, 2006
Description: 9/5 Chapter 3: Recognizing Opportunities: The Role of Cognition Idea generation o Production of ideas for something new Memory, schemas, and prototypes Mental shortcuts and cognitive traps Ideas occur when individuals use existing knowledge they hav...
aem120-9-7
Path: Cornell >> AEM >> 1200 Fall, 2006
Description: 9/7 Chapter 3: Maximizing Opportunity Recognition Creativity o Items or ideas produced are both Novel (original, unexpected) Appropriate or useful Information and opportunity Where is the middle? Concepts o Internal mental structures developed to ...
