tailieunhanh - Answers to Exercises

sequence; like a set but order matters vector spaces vectors zero vector, zero vector of V bases standard basis for Rn basis vectors matrix representing the vector set of n-th degree polynomials set of n×m matrices span of the set S direct sum of subspaces isomorphic spaces homomorphisms, linear maps matrices transformations; maps from a space to itself square matrices matrix representing the map h matrix entry from row i, column j determinant of the matrix T rangespace and nullspace of the map h generalized rangespace and nullspace Lower case Greek alphabet name alpha beta gamma. | Answers to Exercises .dietecvt rffyefaa. Jim Hefferon 1 2 3 1 x 1 2 x. 3 1 6 2 8 1 Notation R N C . . j V W U V w 0 V B D En 1 . . . en RepB V Pn Mn m S M N V W h g H G t s T S ReP B D h h- vi j T R h N h R1 h N1 h real numbers natural numbers 0 1 2 . complex numbers set of . such that . sequence like a set but order matters vector spaces vectors zero vector zero vector of V bases standard basis for Rn basis vectors matrix representing the vector set of n-th degree polynomials set of n X m matrices span of the set S direct sum of subspaces isomorphic spaces homomorphisms linear maps matrices transformations maps from a space to itself square matrices matrix representing the map h matrix entry from row i column j determinant of the matrix T rangespace and nullspace of the map h generalized rangespace and nullspace Lower case Greek alphabet name character name character name character alpha iota L rho p beta p kappa sigma Ơ gamma lambda X tau T delta 5 mu g upsilon À epsilon 6 nu V phi Á zeta c xi chi  eta r omicron o psi à theta e pi omega Cover. This is Cramer s Rule for the system x 2y 6 3x y 8. The size of the first box is the determinant shown the absolute value of the size is the area . The size of the second box is x times that and equals the size of the final box. Hence x is the final determinant divided by the first determinant. These are answers to the exercises in Linear Algebra by J. Hefferon. Corrections or comments are very welcome email to An answer labeled here as for instance matches the question numbered 4 from the first chapter second section and third subsection. The Topics are numbered .