blargoner / math-algebra-greub-errata

Errata from Linear Algebra and Multilinear Algebra by Greub
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Closed chanupedia closed 3 years ago

chanupedia commented 3 years ago

I have the following additions: Linear Algebra: (4, 20): ‘A partially ordered set’ should be ‘a nonempty partially ordered set’. (12, last lines): In the proof of Theorem I, it should be remarked that 𝒜(R,S) is nonempty, since R∈𝒜(R,S). This is necessary for Zorn's lemma to apply. (22, -15): in part (v) of Problem 5, the notion of kernel hasn't been defined either (31, 14): in Problem 7, ‘composition’ should be ‘decomposition’. (31, 19): in Problem 8, a question mark is missing after the second instance of E₁. (41, ??): in Problem 9, σ[₁]should be σ[ₚ₊₁]. This problem doesn't figure in the softcover printing (the one with blue, orange and white cover), but does in the hardcover one (the one with yellow cover). (54, problem 8) the hypothesis that E and F are stable under S_E and S_F, respectively, is missing (62, problem 1) in the hardcover edition, the vertical line in the direct sum symbol is missing. This doesn't happen in the softcover edition. (70, -3): in the hardcover edition, the bar is missing over the letter phi. This doesn't happen in the softcover edition. (75, -2): E/E₁⊥ and F/F₁⊥ should be E/E₁⊥ and F/F₁⊥, respectively. CHANGE "(109, 10): in problem 6, it must be assumed that E is real" to: "(109, 10): in problem 6, it must be assumed that Γ contains no square roots of -1". (132, 12) 'the dimension of n is even' should be ‘the dimension of E is even’ or ‘n is even’. (132, 14) 'orientated' should be 'oriented' (142, -11) ‘isotopices’ should be ‘isotopies’. (227, -3) in the hardcover edition, the codomain of π is written as E, instead of the correct E₁. (371, 11) the last superscript should be kᵣ, not k₂.

MULTILINEAR ALGEBRA (12, Exercise 5b) "×" should be "⊗". The premise implicitly identifies the subspace Δ⊗E_Δ of Γ⊗E_Δ with E_Δ. This is justified by the example in p. 6 together with the result on the compatibility of tensor products and subspaces established in section 1.9.

blargoner commented 3 years ago

@chanupedia Thank you! Which printing of the hardcover are you using? My yellow hardcover doesn't have some of the errors you listed, and doesn't have a problem 9 on page 41.

Here's the copyright page from mine:

chanupedia commented 3 years ago

I'm sorry, I don't have a copy with me now :(

blargoner commented 3 years ago

@chanupedia Thanks again! I made the changes noted below. I also just added errata from Chapter XIII.

I have the following additions: Linear Algebra: (4, 20): ‘A partially ordered set’ should be ‘a nonempty partially ordered set’.

Not added. The empty subset is a chain, which has an upper bound by assumption, so the partially ordered set must be nonempty.

(12, last lines): In the proof of Theorem I, it should be remarked that 𝒜(R,S) is nonempty, since R∈𝒜(R,S). This is necessary for Zorn's lemma to apply.

Added (with changes).

(22, -15): in part (v) of Problem 5, the notion of kernel hasn't been defined either

Added.

(31, 14): in Problem 7, ‘composition’ should be ‘decomposition’.

Added.

(31, 19): in Problem 8, a question mark is missing after the second instance of E₁.

Added.

(41, ??): in Problem 9, σ[₁]should be σ[ₚ₊₁]. This problem doesn't figure in the softcover printing (the one with blue, orange and white cover), but does in the hardcover one (the one with yellow cover).

Not added because it doesn't appear in my copy.

(54, problem 8) the hypothesis that E and F are stable under S_E and S_F, respectively, is missing

Not added because it's obvious, and Greub indicates as much in the definition of "irreducible" immediately prior ("...if the only stable subspaces are F=0 and F=E.")

(62, problem 1) in the hardcover edition, the vertical line in the direct sum symbol is missing. This doesn't happen in the softcover edition.

Not added because it doesn't appear in my copy.

(70, -3): in the hardcover edition, the bar is missing over the letter phi. This doesn't happen in the softcover edition.

Not added because it doesn't appear in my copy.

(75, -2): E/E₁⊥ and F/F₁⊥ should be E/E₁⊥ and F/F₁⊥, respectively.

Added.

CHANGE "(109, 10): in problem 6, it must be assumed that E is real" to: "(109, 10): in problem 6, it must be assumed that Γ contains no square roots of -1".

Changed "must" to "should". 😃

(132, 12) 'the dimension of n is even' should be ‘the dimension of E is even’ or ‘n is even’.

Added.

(132, 14) 'orientated' should be 'oriented'

Added.

(142, -11) ‘isotopices’ should be ‘isotopies’.

Added.

(227, -3) in the hardcover edition, the codomain of π is written as E, instead of the correct E₁.

Not added because it doesn't appear in my copy.

(371, 11) the last superscript should be kᵣ, not k₂.

Not added because I'm not covering this section (but the error does appear in my copy).

MULTILINEAR ALGEBRA (12, Exercise 5b) "×" should be "⊗". The premise implicitly identifies the subspace Δ⊗E_Δ of Γ⊗E_Δ with E_Δ. This is justified by the example in p. 6 together with the result on the compatibility of tensor products and subspaces established in section 1.9.

Not added. Your remark about the identification is correct, and implies that Δ×E_Δ is a subset of Γ×(Γ⊗E_Δ) to which the scalar multiplication defined on Γ×(Γ⊗E_Δ) can be restricted as Greub intends.