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# DiffEquationsLinearAlg MATH0033

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This 6 page Class Notes was uploaded by Aileen Davis on Tuesday October 20, 2015. The Class Notes belongs to MATH0033 at Sierra College taught by DebraHill in Fall. Since its upload, it has received 77 views. For similar materials see /class/225374/math0033-sierra-college in Mathematics (M) at Sierra College.

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Date Created: 10/20/15

Section 21 g 117 De nitions symmetric matrix skewsymmetric matrix TrueFalse l A diagonal matrix is both upper triangular and lower triangular 3 If A is a symmetric matrix then so is AT 5 A skewsymmetric matrix must have zeros along the main diagonal Section 22 g 129 De nitions scalar multiplication identity matrix Kronecker delta symbol TrueFalse 5 For ngtltn matricesA andB we have AB2 2A2 2ABB2 Section 23 g 138 De nitions homogeneous system consistent system inconsistent system augmented matrix TrueFalse 1 If a linear system of equations has an m gtltn augmented matrix then the system has In equations and n unknowns Section 24 g 148 De nitions elementary row operations rowequivalent matrices row echelon form rank of a matrix reduced rowechelon matrix TrueFalse l A matrixA can have many rowechelon forms but only one reduced rowechelon form Section 25 g 158 De nitions Gaussian elimination GaussJordan elimination free variables bound variables trivial solution TrueFalse 3 For a linear system Ax b every column of the rowechelon form of A corresponds to either a bound variable or a free variable but not both of the linear system 5 A linear system is consistent iff there are free variables in the row echelon fOI H l of the corresponding augmented matrix Section 2 6 g 169 De nitions inverse invertible singular nonsingular TrueFalse 3 A linear system AX b with an n gtltn invertible coef cient matrixA has a unique solution 9 HA is a 5X5 matrix of rank 4 thenA is not invertible Section 2 7 g I 79 De nitions elementary matrix LU Factorization of a matrix TrueFalse 1 Every elementary matrix is invertible 3 Every matrix can be expressed as a product of elementary matrices 5 If is a permutation matrix then P PU Section 2 8 g 182 TrueFalse 3 HA is a 3X 3 matrix with rankA 2 then the linear system AX b must have infinitely many solutions Section 31 g 197 De nitions determinant TrueFalse 1 If A is a 2 X2 lower triangular matrix then detA is the product of the elements on the main diagonal of A 5 HA andB are 2X2 matrices then detAB detA detB 7 A matrix containing a row of zeros must have zero determinant Section 32 Q 209 TrueFalse 1 If each element of an n gtltn matrix is doubled then the determinant of the matrix also doubles 2 5 The matrix x2 x is not invertible iff x 0 or y 0 y y Section 33 g 221 De nitions minor cofactor matrix of cofactors adjoint Cramer s rule TrueFalse l The 23 minor of a matrix is the same as the 23cofactor of the matrix 3 Cofactor expansion of a matrix along any row or column will yield the same result Section 41 g 239 De nitions vectors in Rquot vector addition scalar multiplication zero vector additive inverse components of a vector TrueFalse l The vector xy in R2 is the same as the vector xy0 in R3 3 The solution set to a linear system of 4 equations and 6 unknowns consists of a collection of vectors in R6 9 If x is a vector in the rst quadrant of R2 then any scalar multiple kx of x is still a vector in the first quadrant of R2 11 Three vectors X y and z in R3 always determine a 3dimensional solid region in R3 Section 42 g 248 De nitions vector space real or complex closure under addition closure under scalar multiplication commutativity of addition associativity of addition existence of zero vector existence of additive inverses unit property associativity of scalar multiplication distributive properties TrueFalse l The zero vector in a vector space V is unique 3 The set Z of integers together with the usual operations of addition and scalar multiplication forms a vector space The additive inverse of a vector v in a vector space Vis unique 7 The set 01 with the usually operations of addition and scalar U multiplication forms a vector space Section 43 g 256 De nitions subspace trivial subspace null space of a matriXA TrueFalse 3 The points in R2 that lie on the line y mx b fOI H l a subspace of R2 iff b 0 A nonempty set S of a vector space Vthat is closed under scalar multiplication contains the zero vector of V If V R3 and S consists of all points on the xy plane the xz plane and the yz plane then S is a subspace of V Section 44 g 265 De nitions linear combination linear span spanning set TrueFalse l 3 7 11 The linear span of a set of vectors in a vector space Vforms a subspace of V If S is a spanning set for a vector space Vand W is a subspace of V then S is a spanning set for W Every vector space Vhas a finite spanning set If m lt n then any spanning set for Rquot must contain more vectors than any spanning set for Rquot Section 4 5 g 278 De nitions linearly dependent set linear dependency linear independent set minimal spanning set Wronskian of a set of functions TrueFalse 1 Every vector space Vpossesses a unique minimal spanning set 5 If the Wronskian of a set of functions is nonzero at some point x0 in an interval I then the set of functions is linearly independent 9 If the Wronskian of a set of functions is identically zero at every point of an interval I then the set of functions is linearly dependent Section 4 6 g 290 De nitions basis standard basis nitedimensional dimension extension of a subspace basis TrueFalse l A basis for a vector space Vis a set S of vectors that spans V 3 A vector space Vcan have many different bases 5 If Vis an ndimensional vector space then any set S of m vectors with mgtn must span V 7 Two vectors in P3 must be inearly independent 9 If V is an ndimensional vector space then every set S with fewer than n vectors can be extended to a basis for V Section 4 7 Q 299 De nitions ordered basis components of a vector relative to an ordered basis changeofbasis matrix TrueFalse 1 Every vector in a nitedimensional vector space V can be expressed uniquely as a linear combination of vectors comprising a basis for V 3 A changeofbasis matrix is always a square matrix no problems from Section 48 no problems from Section 49 Section 4 10 g 312 TrueFalse l The set of all row vectors of an invertible matrix is linearly independent Section 411 g 321 De nitions inner product axioms of an inner product real complex inner product space norm angle CauchySchwarz inequality TrueFalse 1 If v and W are linearly independent vectors in an inner product space V then ltvwgt 0 5 In any vector space V there is at most one valid inner product lt gt that can be defined on V 7 If p x a0 alx azx2 and qx 2 0 blx bzx2 then we can de ne an inner product on P2 via ltpqgt aobo Section 412 g 331 De nitions orthogonal vectors orthogonal set unit vector orthonormal vectors orthonormal set normalization orthogonal basis orthonormal basis GramSchmidt process orthogonal projection TrueFalse 1 Every orthonormal basis for an inner product space Vis also an orthogonal basis for V

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