Mathematics & Statistics Texas Tech University Kent Pearce Department of Mathematics and Statistics Texas Tech University Lubbock, Texas 79409-1042 Voice: (806)742-2566 x 226 FAX: (806)742-1112 Email: kent.pearce@ttu.edu

 Math 2360 Linear Algebra Fall 2012 Larson, Ron Linear Algebra Cengage

Review Exam III
 Section Content Suggested Problems Section 4.7 Coordinate of a vector with respect to a basis Coordinate matrix Notation [x]B Change of basis in Rn Transition matrix from basis B' to B Inverse of a transition matrix Theorem 4.21 Transition matrix from basis B to B' Coordinate representation in general vector space V Pages 210-211 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 35, 37 Section 4.8 Linear differential equation Wronskian of a set of functions Wronskian test for linear independence Pages 219-223 13, 14, 17, 20, 21 Section 5.1 Length or norm of a vector in Rn Length of a scalar multiple Unit vector Distance between two vectors in Rn Dot product of two vectors in Rn Properties of the dot product Cauchy-Schwarz inequality Angle between two non-zero vectors in Rn via dot product Definition of orthogonal Triangle Inequality in Rn Pythagorean Theorem in Rn Pages 235-236 7-8, 11-12, 20-21, 25-26, 41-42, 48-51 Section 5.2 Definition of inner product on a vector space Examples of inner products for Rn, Mm,n, C[a,b] Properties of inner products Definitions of length, distance and angle Cauchy-Schwarz inequality Triangle Inequality Pythagorean Theorem Orthogonal Projection of a vector u on to a vector v Pages 245-247 17, 19, 21, 23, 29, 35, 39, 41, 47-49, 73, 75, 78-79 Section 5.3 Orthogonal sets Orthonormal sets Orthogonal basis, Orthonormal basis Theorem 5.10 Orthogonal sets is linearly independent Corollary to Theorem 5.10, Page 251 Theorem 5.11 Coordinates of vector with respect to an orthogonal basis Theorem 5.12 Gram-Schmidt Orthogonalization Process Pages 257-258 1, 4, 7, 10, 13, 21, 25, 27, 31, 35, 39, 41 Section 5.4 Least Square Regression Line Least Squares Problem Orthogonal Subspaces Orthogonal Complement Direct Sum Properties of Orthogonal Subspaces Projection onto a Subspace which is the span of an orthogonal basis Theorem 5.15 Orthogonal Projection and Distance Fundamental subspaces of a matrix Theorem 5.16 Fundamental Subspaces of a Matrix Solution of the Least Squares Problem Normal equation of the least squares problem Orthogonal projections onto a subspace Pages 269-270 5-7, 15-17, 19-21, 23-25, 29-31, 33-35 Section 5.5 Least squares approximations in calculus Definition of least squares approximation Theorem 5.19 Least squares approximating function Pages 282-283 63-65, 69-71

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