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 2009 Leon, Steven J Linear Algebra Pearson

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 three types of solutions sets solution sets with cardinality one (unique solution) solution sets with infinite cardinality (infinitely many solutions) solution sets with cardinality zero (no solutions -- inconsistent) equivalent systems of m linear equations in n variables strict triangular system of m linear equations in n variables back substitution coefficient matrix of 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 pivotal row, pivotal element 1, 5, 6 Section 1.2 row echelon form Gaussian elimination types of systems overdetermined undetermined reduced row echelon form application #1: traffic flow 1, 2, 3, 5, 7, 8, 13, Section 1.3 matrix notation vectors definition of equality for two matrices definition of sum for two matrices definition of scalar multiplication for a scalar and a matrix algebraic properties of matrix addition definition of matrix product for a row vector and a column vector definition of product for two matrics 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 powers of matrices (nxn) matrix equation representation of a system of m linear equations in n variables definition of a linear combination Consistency Theorem for Systems of Linear Equations (pg 37) invertible (non-singular) matrix Theorem 1.3.3 (pg 49) transpose of a matrix algebraic properties of transposes symetric matrix 1, 2, 4, 10, 13, 22, 23, 25 Section 1.4 elementary matrices 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 row equivalance of matrices Theorem Equivalent conditions for Non-singularity (pg 65) construction of matrix inverse via row reduction of the maxtrix augmented with the identity triangular matrices diagonal matrices Theorem: Triangular matrix is non-singular if trace is non-zero 1, 3, 4, 5, 6, 10, 13, 23 Section 2.1 determinant of 2x2 matrix definition of the determinant of a matrix Theorem 2.1.1 (pg 95)properties of determinants determinant of AT determinant of triangular matrix determinant of A if A has a zero row (column) determinant of A if A has two identical rows (columns) 1, 2, 3, 5-6, 11 Section 2.2 Theorem 2.2.1 (pg 98) determinants of elementary matrices E and determinants of multiplication (on the left) by E type I: det(E) = -1; det(EA) = -det(A) type II: det(E) = alpha; det(EA) = alpha det(A) type III: det(E) = 1; det(EA) = det(A) Theorem 2.2.2 A is non-singular iff det(A) not equal 0 computation (simplificiations) of determinants via reductions (eliminations) by elementary matrix operations Theorem 2.2.3 det(AB) = det(A) det(B) 2a, 3, 4, 6, 7, 9 Section 2.3 adjoint of an nxn matrix Cramer's Rule 1, 2, 6

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