Linear Algebra. Answers To Exercises

Answers to Exercises Linear Algebra Jim Hefferon ¡ 1¢ 3 ¡2¢ 1 ¯ ¯1 ¯ ¯3 ¯ 2¯¯ 1¯ x1 · ¡ 1¢ 3 ¡2¢ 1 ¯ ¯ ¯x · 1 2 ¯ ¯ ¯ ¯x · 3 1 ¯ ¡6¢ 8 ¡2¢ 1 ¯ ¯ ¯6 2 ¯ ¯ ¯ ¯8 1 ¯ Notation R, R+ , Rn N C ¯ {. . . ¯ . . .} (a .. b), [a .. b] 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, 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 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 delta epsilon zeta eta theta character α β γ δ ² ζ η θ name iota kappa lambda mu nu xi omicron pi character ι κ λ µ ν ξ o π name rho sigma tau upsilon phi chi psi omega character ρ σ τ υ φ χ ψ ω 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, x1 is the final determinant divided by the first determinant. These are answers to the exercises in Linear Algebra by