A Short Linear Algebra Book (answers)

Answers to Exercises from Linear Algebra ¯ ¯1 ¯ ¯3 ¯ 2¯¯ 1¯ ¯ ¯x · 1 ¯ ¯x · 3 ¯ 2¯¯ 1¯ ¯ ¯ ¯6 2 ¯ ¯ ¯ ¯8 1 ¯ Jim Hefferon Notation R N ¯ C {. . . ¯ . . . } h. . . i V, W, U ~v , w ~ ~0, ~0V B, D En = h~e1 , . . . , ~en i ~ ~δ β, RepB (~v ) Pn Mn×m [S] M ⊕N V ∼ =W h, g H, G t, s T, S RepB,D (h) hi,j |T | R(h), N (h) R∞ (h), N∞ (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×m matrices span of the set S direct sum of subspaces isomorphic spaces homomorphisms 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 delta epsilon zeta eta theta symbol α β γ δ ² ζ η θ name iota kappa lambda mu nu xi omicron pi symbol ι κ λ µ ν ξ o π name rho sigma tau upsilon phi chi psi omega symbol ρ σ τ υ φ χ ψ ω Cover. This is Cramer’s Rule applied to the system x + 2y = 6, 3x + y = 8. The area of the first box is the determinant shown. The area of the second box is x times that, and equals the area 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 jimjoshua.smcvt.edu An answer labeled here as, for instance, 1.II.3.4, matches the question numbered 4 from the first chapter, second section, and third subsection. The Topics are numbered separa