Linear Algebra Jim Hefferon
Answers to Exercises Linear Algebra Jim Hefferon ¡ ¢ 1 3 ¡ ¢ 2 1 ¯ ¯ ¯ ¯ ¯1 2¯ ¯3 1¯ ¡ ¢ 1 x · 1 3 ¡ ¢ 2 1 ¯ ¯ ¯ ¯ ¯x · 1 2¯ ¯ ¯ x · 3 1 ¡ ¢ 6 8 ¡ ¢ 2 1 ¯ ¯ ¯ ¯ ¯6 2¯ ¯8 1¯ Notation R real numbers N natural numbers: {0, 1, 2,...} ¯ C complex numbers {... ¯ ...} set of . such that . h...i sequence; like a set but order matters V, W, U vector spaces ~v, ~w vectors ~0, ~0V zero vector, zero vector of V B, D bases n En = h~e1, . , ~eni standard basis for R β,~ ~δ basis vectors RepB(~v) matrix representing the vector Pn set of n-th degree polynomials Mn×m set of n×m matrices [S] span of the set S M ⊕ N direct sum of subspaces V =∼ W isomorphic spaces h, g homomorphisms, linear maps H, G matrices t, s transformations; maps from a space to itself T,S square matrices RepB,D(h) matrix representing the map h hi,j matrix entry from row i, column j |T | determinant of the matrix T R(h), N (h) rangespace and nullspace of the map h R∞(h), N∞(h) generalized rangespace and nullspace Lower case Greek alphabet name character name character name character alpha α iota ι rho ρ beta β kappa κ sigma σ gamma γ lambda λ tau τ delta δ mu µ upsilon υ epsilon ² nu ν phi φ zeta ζ xi ξ chi χ eta η omicron o psi ψ theta θ pi π omega ω Cover. This is Cramer’s Rule for the system x1 + 2x2 = 6, 3x1 + x2 = 8.
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