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# Linear Algebra and Applications MATH 33A

UCLA

GPA 3.55

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This 2 page Class Notes was uploaded by Kaylin Wehner on Friday September 4, 2015. The Class Notes belongs to MATH 33A at University of California - Los Angeles taught by Staff in Fall. Since its upload, it has received 125 views. For similar materials see /class/177820/math-33a-university-of-california-los-angeles in Mathematics (M) at University of California - Los Angeles.

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Date Created: 09/04/15

Math 33aiReview for Midterm 2 34Coordinates o For a basis B go back and forth between a vector i and its B coordinate vector using the formulas 32 SMB and 5 5 13 341 o For a basis B nd the B matrix of a linear transformation T 343 o For a basis B go back and forth between the standard matrix A of a linear transformation T and the B matrix of T using the formulas A STBS 1 and Tlg S lAS 344 0 Know the de nition of similarity of matrices 345 5170rthogonal Projections and Orthonorrnal Bases 0 Know the formula for orthogonal projection of a vector i onto a subspace V of R with orthonormal basis 73117n 515 0 Know the de nition and properties of the orthogonal complement VL of a subspace V 517 518 0 Be able to prove and use the Pythagorean Theorem 519 c Find the angle 9 between two vectors 5112 527GramSchmidt Orthogonalization and QR Factorization 0 Apply the Gram Schmidt process to a basis 171 17m of a subspace V of R to nd an orthonormal basis 731 7Zm of V 521 c Find the QR factorization of an n x m matrix M with linearly independent columns 522 5370rthogonal Transformations and Orthogonal Matrices o A transformation is orthogonal if it preserves length of vectors 531 It then also pre serves dot products and angles between vectors and in particular orthogonality 532 An important alternate de nition of orthogonal transformations and matrices is in terms of orthonormal bases 533 0 Know de nition of transpose of a matrix symmetric matrix skew symmetric matrix 535 0 An 71 x 71 matrix is orthogonal if and only if ATA In 537 0 Know how transposes interact with products inverses and rank 539 0 Given an orthonormal basis 731 7Zm for a subspace V of R the matrix of orthogonal projection onto V is QQT where Q has the 73 in its columns 5310 547Least Squares and Data Fitting 0 Understand the meaning of a least squares solution of a linear system Ai 6 544 c Find the least squares solutions of a linear system AE I by solving the consistent system ATAE ATE 545 546 0 Given any basis 171 17m for a subspace V of R the matrix of orthogonal projection onto V is AATA 1AT where A has the 17 in its columns 547 0 Fit a curve of a certain kind to a set of data 61iIntr0duction t0 Determinants Compute the determinant of an n x 71 matrix by Laplace expansion along a row or a column 651 The determinant of an upper or lower triangular matrix is the product of its diagonal entries 616 627Pr0perties 0f the Determinant o A square matrix A is invertible if and only if detA 344 0 622 0 Use Gauss Jordan elimination to compute the determinant of a matrix A 623 0 Use the formula detAB detA detB to deduce results on the determinants of similar matrices an inverse matrix a transpose orthogonal matrices etc 624 5 6 7 631 637Ge0metrical Interpretations of the Determinant Cramer7s Rule o If A is an n x 71 matrix with columns 17117n then ldetAl llalllll zill ll ill where are de ned as in the Gram Schmidt process 634 o The m volume of the m parallelepiped de ned by the vectors 171 17m in R is detATA where A has 171 17m in its columns If m n this volume is detA 637

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