tailieunhanh - Answers to Exercises Linear Algebra

Linear algebra is the branch of mathematics concerning vector spaces, often finite or countably infinite dimensional, as well as linear mappings between such spaces. Such an investigation is initially motivated by a system of linear equations in several unknowns. Such equations are naturally represented using the formalism of matrices and vectors | Answers to Exercises Linear Algebra Jim Hefferon 1 2 3 1 x 1 2 x 3 1 6 2 8 1 Notation R R R N C . I . a . b a . b __ V W U v lb 0 0v B D E ei . en ữ Reps v P Mnxm s M N V W h g H G t s T s RePSjp h h - A vi j T R h N h M NJh real numbers reals greater than 0 n-tuples of reals natural numbers 0 1 2 . complex numbers set of . such that . interval open or closed of reals between a and b sequence like a set but order matters vector spaces vectors zero vector zero vector of V bases standard basis for R 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 alpha a iota I rho beta 3 kappa K sigma gamma Y lambda X tau delta 5 mu p upsilon epsilon e nu V phi zeta z xi chi eta n omicron o psi theta e pi n omega character p Ơ T V Ộ X ý w Cover. This is Cramer s Rule for the system X1 2x2 6 3x1 x2 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 X1 times that and equals the size of the final box. Hence Xi 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 .