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 I
 Section Content Suggested Problems Section 1.1 linear equation in n variables system of m linear equations in n variables solution of system of m linear equations in n variables solution set of system of m linear equations in n variables parameteric representation of solutions consistent and inconsistent systems of linear equations three types of solutions sets for systems of linear equations solution sets with cardinality one (unique solution) solution sets with infinite cardinality (infinitely many solutions) solution sets with cardinality zero (no solutions -- inconsistent) back substitution equivalent systems of m linear equations in n variables operations that produce equivalent systems interchange two equations multiply an equation by a non-zero number add a multiple of one equation to another equation Pages 10-12 1-4, 7, 9, 11-17, 27-30, 37-38, 47-48, 53-54, 67-68 Section 1.2 matrix notation coefficient matrix of a system of m linear equations in n variables augmented matrix for a system of m linear equations in n variables elementary row operations interchange two rows multiply a row by a non-zero number add a multiple of one row to a second row and replace the second row row echelon form and reduced row echelon form Gaussian elimination with back substituion Gauss-Jordan elimination homoegeous systems of linear equations always consistent Pages 22-24 7-10, 11-14, 19-24, 25-27, 31-32, 35-36, 41-43 Section 1.3 polynomial curve fitting translating large x-values network analysis electric circuit analysis Pages 32-34 3, 5, 7, 18, 27-28, 31, 33 Section 2.1 definition of equality for two matrices definition of sum for two matrices definition of scalar multiplication for a scalar and a matrix definition of matrix product for a row vector and a column vector definition of product for two matrics matrix equation representation of a system of m linear equations in n variables solve a system of linear equations via matrix representation definition of a linear combination of column vectors Pages 48-51 3-4, 7-8, 11, 15, 17, 21-22, 25-26, 31-38, 41-42, 45-46, 49-50 Section 2.2 algebraic properties of matrix addition and scalar multiplication algebraic properties of matrix multiplication differences between algebraic properties of real multiplication and matrix multipication non-commutativity of multiplication non-cancellation of multiplication in equations three types of solutions sets for systems of linear equations solution sets with cardinality one (unique solution) solution sets with infinite cardinality (infinitely many solutions) solution sets with cardinality zero (no solutions -- inconsistent) transpose of a matrix algebraic properties of transposes Pages 59-61 1, 3, 5, 7, 9, 11, 23, 25, 33-34, 37, 39, 43 Section 2.3 definition of the inverse of a square matrix uniqueness of the inverse of a matrix finding the inverse of a matrix by Gauss-Jordan elimination algebraic properties of the inverse of a matrix solving a system of equations using the inverse of a matrix Pages 71-73 3, 5, 7, 9, 12-13, 15, 23-24, 31-33, 41, 45, 47 Section 2.4 definiton of an elementary matrix multiplication by elementary matrices represents elementary row operations type I: interchanges two rows upon multiplication (on the left) type II: multiplies a row by a non-zero constant upon multiplication (on the left) type III: adds a multiple of one row to a second row and replaces the second row with the sum upon multiplication (on the left) multiplication of elementary matrix on the right induces action on the columns definition of row equivalance of matrices elementary matrices are invertible Theorem 2.15 on qquivalent conditions for Non-singularity (pg 78) definition of LU factorization of a matric construction of LU factorization via (elementary) row reduction solving systems of linear equations via LU factorizations Pages 82-83 1-8, 9-12, 13-14, 17-20, 27, 31, 35-36, 41, 43, 45

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