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# MATRIX ALGEBRA & APPLS MA 322

UK

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This 5 page Class Notes was uploaded by Kennith Herman on Friday October 23, 2015. The Class Notes belongs to MA 322 at University of Kentucky taught by Joshua Roberts in Fall. Since its upload, it has received 6 views. For similar materials see /class/228127/ma-322-university-of-kentucky in Mathematics (M) at University of Kentucky.

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Date Created: 10/23/15

00N CnggtcopHH MA 322021 2nd Summer Session Review for Final Exam De nitions Solution set Augmented matrix ConsistentInconsistent systems Row equivalent Reduced Echelon form PivotPivot columnPivot position Parametric form of solutions Vector scalar vector addition scalar multiplication columnrow vector Linear combination Spanspanning set Matrix equationvector equationlinear system Matrix additionmultiplicationinversionequalitytransposesimilaritydiagonal Homogeneous and non homogeneous systems Parametric vector form of a line and plane Linear independence linear dependence Linear dependence relation Linear transformation Onto amp one to one domain codomain range Matrix transformations Determinant D D D 4 3 CT 3 p 3 3 H 3 3 3 35 3 3 I 3 3 4 O 4 H 4 4 C40 4 4 0 F 9 7 03 00 p E0 Elementary matrix Triangular square symmetric and diagonal matrices Vector space and subspace Null space and column space Basis Dimension of a vector space Rank Eigenvalue eigenvector eigenspace Characteristic equation and polynomial lnner product of two vectors Length or norm of a vector distance between two vectors Orthogonal vectors Wl Orthogonal set Orthogonal basis Orthogonal project of a vector onto a line Orthogonal projection of a vector onto a subspace Orthonormal set Orthonormal basis A least squares solution of Am b The quadratic form wTAw Positive de nite negative de nite inde nite quadratic forms II Theorems and other things you should know how to do H Sovle a system of equation with row reduction E0 Determine existenceuniqueness about systems 9 Vector and matrix algebra 7 Explain the parallelogram law for addition of vectors 9 Use row reduction to express a vector and a linear combination of other vector CT Determine the span of a set of vectors 5 Solve the homogeneous equation Aw 0 and the non homogeneous equation Aw b 00 Answer existenceuniqueness by looking at an echelon form of a matrix 3 Statements of Theorem 4 page 43 10 Write a solution in parametric vector form 1 Find linear dependence relations among a group of vectors 12 Determine if a set of vectors is linear independent 13 Find the matrix of a linear transformation 14 Show that a given function is a linear transformation 5 Apply theorems 11 and 127 pgs 887 89 16 aml description of a matrix 17 Properties of the transpose of a matrix 18 How to nd the inverse of a square matrix 19 How to apply the inverse of a matrix to solve Aw b and answer existence uniqueness questions 20 The inverse of a product of matrices and of the transpose of a matrix 1 The statements ofthe lnvertible Matrix Theorem all statements7 not just those in Chapter 2 and how to apply them to problems F 9 7 9 03 00 p O H F 9 7 Cf 03 5 00 0 H F 9 4 LU Factorization Find the determinant of a matrix Properties of determinants Given a vector space V and a subset U of V determine whether or not U is a subspace of V Examples of vector spaces other than R Find bases for null space and column space Determine whether or not a given set is a basis for a subspace Apply the Spanning Set Theorem Theorems 9 10 11 and 12 12 is the Basis Theorem from 45 pages 256 259 Find the dimension of the column space and null space of a matrix A The rank of a matrix The Rank Theorem Find the eigenvalues eigenvectors and a basis for eigenspaces of a matrix A The eigenvalues of a diagonal matrix occur on the diagonal Using the characteristic equation Diagonalize a matrix The Diagonalization Theorem lnner product algebra Find the length and a vector nding the distance between two vectors Decide if two or more vectors are orthogonal The Pythagorean Theorem for R Given a subspace W of R decide if a vector is in lVL Calculate the orthogonal projection of a vector y onto a vector U or a subspace W of R 4 46 4 4 00 4 5 U 5 p 0 Given 3 and u7 express an as the sum of two vectorsione is in Span u and the other orthogonal to u Do the same7 but for a subspace W instead of a line Find the distance between a vector and a line and between a vector and a subspace W The Best Approximation Theorem Solving the Least Squares problem Calculate and classify quadratic forms positive de nite7 etc

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