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# Advanced Calculus MATH 567

WVU

GPA 3.8

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This 2 page Class Notes was uploaded by Rae Kutch on Saturday September 12, 2015. The Class Notes belongs to MATH 567 at West Virginia University taught by Staff in Fall. Since its upload, it has received 31 views. For similar materials see /class/202659/math-567-west-virginia-university in Mathematics (M) at West Virginia University.

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Date Created: 09/12/15

Review topics Vectors vector addition scalar multiplication Linear combinations Matrix algebra addition multiplication properties Interpretations of matrix multiplication eg Each column of AB is a combination of cols of A generated by corresponding column of B Each row of AB is a combination of rows of B generated by corresponding row of A An element of AB is a row ofA times a column of B Subspaces of Rn 7 what is a subspace how do we recognize a subspace Space spanned by a set of vectors 7 matrix form of such a space all vectors Ax same as column space of A same as space spanned by columns of A Space de ned by homogeneous conditions All solutions of Ax0 same as null space of The dimension ofa subspace basis ofa subspace Row echelon and reduced row echelon form of a matrix 7 row operations Four subspaces Column space row space null space left null space null space of AT Determining a basis for any of the subspaces using elimination The rank of a matrix and its quot to the J39 39 of the 39 The general solution of Axb homogeneous plus particular Determining linear independence of a set of vectors nding linear dependencies among vectors Inverses 7 calculating inverses using elimination using inverses in linear equations inverse of a product of square matrices Singularnonsingular matrices Determinants Basic properties and derived properties The big formula Effect of rowcolumn operations on the determinant Laplace expansion formula using minors Evaluating a determinant using row operations and expansion Determinant of a triangular matrix Determinant of A What a determinant determines Formula for the inverse using a determinant Cramer s rule 1 Singular 7 not invertible definition 7 detA0 7 columns dependent 7 rows dependent 7 rankAltn 7 Ax0 has a nonzero solution 7 A not row equivalent to I DetABdetAdetB for square matrices only Review topics for exams Solving rst order DE s Linear separable Plotting simple integral curves Determining where an initial value problem has a unique solution either with an explicit solution for a nonlinear de or in general for a linear de Second order linear de s General principles lnitial value problems where do solutions exist Homogeneous de s superposition of of solutions independent solutions and the Wronskian general solution of a homogeneous 2nd order DE Second order nonhomogeneous de s Homogeneous particular General formula for a particular solution variation of parameters formula yp y1lyWy2lyW Constant coef cient DE s Exponential solutions characteristic equation determination of general solution based on roots ofthe characteristic equation Undetermined coefficients form of the solution for constant coeff de s with special nonhomogeneous terms polynomialexponentialsincos

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