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Corollary 10.58. Let \( C \) be an irreducible projective curve with function field \( \mathbb{K} = \mathbb{k}\left( C\right) \), let \( P \) be a point of \( C \), and let \( {M}_{P} \) be the maximal ideal of \( {\mathcal{O}}_{P}\left( C\right) \) . Then there exists a discrete valuation \( v \) of \( \mathbb{K} \) d... | Proof. Without loss of generality, we may assume that \( C \) is affine. Let \( {\mathfrak{m}}_{P} \) be the maximal ideal in the affine coordinate ring \( A\left( C\right) \) consisting of all functions vanishing at \( P \), and let \( S \) be the set-theoretic complement of \( {\mathfrak{m}}_{P} \) in \( A\left( C\ri... | Yes |
Corollary 10.59. If \( \mathbb{K} \) is a function field in one variable over \( \mathbb{k} \) and if \( v \) is a discrete valuation of \( \mathbb{K} \) defined over \( \mathbb{k} \) with valuation ring \( {R}_{v} \), then there exists an irreducible nonsingular affine curve \( C \) over \( \mathbb{k} \) with function... | Proof. Choose an element \( x \) of \( \mathbb{K} \) such that \( v\left( x\right) > 0 \) . Define \( R = \mathbb{k}\left\lbrack x\right\rbrack \) . Since \( v\left( x\right) \neq 0, x \) is transcendental over \( \mathbb{k} \), and \( \mathbb{K} \) is a finite algebraic extension of the field of fractions \( \mathbb{k... | Yes |
Let \( C \) be the irreducible nonsingular affine curve constructed in Corollary 10.59 and having function field \( \mathbb{K} = \mathbb{k}\left( C\right) \), and regard \( C \) as a subvariety of its projective closure \( \bar{C} \) . Then there are only finitely many discrete valuations \( {v}^{\prime } \) of \( \mat... | Proof. We go over the argument in Corollary 10.59 with the same element \( x \) and with any discrete valuation \( {v}^{\prime } \) defined over \( \mathbb{k} \) such that \( {v}^{\prime }\left( x\right) \geq 0 \) . This inequality implies that \( {v}^{\prime } \) is \( \geq 0 \) on \( \mathbb{k}\left\lbrack x\right\rb... | Yes |
Lemma 10.61. Let \( R \) be a Noetherian integrally closed domain with field of fractions \( F \), let \( K \) be a finite separable extension of \( F \), and let \( T \) be the integral closure of \( R \) in \( K \) . Then \( T \) is Noetherian and is finitely generated as an \( R \) module. | Proof. In effect, this result was proved in Basic Algebra. In more detail: With the above assumptions and also the assumption that every nonzero prime ideal of \( R \) is maximal (i.e., that \( R \) is a Dedekind domain), the proof of Theorem 8.54 of Basic Algebra showed that \( T \) is a Dedekind domain. The hard part... | Yes |
Lemma 10.62 (Noether Normalization Lemma). Let \( k \) be an infinite field, let \( R = k\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) be a finitely generated integral domain over \( k \), and let \( K = k\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) be the field of fractions of \( k \) . Then for a suitable \( d... | Proof. Let \( I \) be the kernel of the quotient homomorphism \( k\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \rightarrow k\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) . The core of the proof involves a single nonzero \( f \) in \( I \) . The idea is to replace \( {X}_{1},\ldots ,{X}_{n - 1} \) by new... | Yes |
Theorem 10.63. Every birational equivalence class of irreducible projective curves contains a nonsingular such curve, and this curve is unique within the equivalence class up to isomorphism of varieties. Any irreducible nonsingular quasiprojective curve is isomorphic to an open subvariety of some irreducible nonsingula... | Proof of THEOREM 10.63. Let \( \mathbb{K} \) be the given function field, and let \( {C}_{1},\ldots ,{C}_{m} \) be the irreducible nonsingular affine curves described two paragraphs before this paragraph. In each case the function field of the curve is isomorphic to \( \mathbb{K} \) by some fixed isomorphism, but we sh... | No |
Proposition 10.64. If \( \mathfrak{a} \) is a proper monomial ideal in \( \mathbb{k}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \), then\n\n(a) the vector subspace \( V\left( \left\{ {{X}_{i} \mid i \notin \left\{ {{j}_{1},\ldots ,{j}_{k}}\right\} }\right\} \right) \) is contained in \( V\left( \mathfrak{a}\rig... | Proof. For (a), first suppose that \( V\left( \left\{ {{X}_{i} \mid i \notin \left\{ {{j}_{1},\ldots ,{j}_{k}}\right\} }\right\} \right) \) is contained in \( V\left( \mathfrak{a}\right) \), and suppose that \( \alpha \) is in \( \left\langle {{e}_{{j}_{1}},\ldots ,{e}_{{j}_{k}}}\right\rangle \) . Let \( P = \left( {{x... | Yes |
Theorem 10.65. If \( \mathfrak{a} \) is a proper monomial ideal in \( \mathbb{k}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \), then the complementary set \( \mathcal{C}\left( \mathfrak{a}\right) \) of monomials is a disjoint union\n\n\[ \mathcal{C}\left( \mathfrak{a}\right) = {C}_{0} \cup \cdots \cup {C}_{n} \... | Proof. We proceed by induction on \( n \), and we may assume that \( \mathfrak{a} \neq 0 \) . The example above shows for \( n = 1 \) that \( \mathcal{C}\left( \mathfrak{a}\right) \) is a finite set if \( \mathfrak{a} \) is a nonzero proper ideal. Thus \( \mathcal{C}\left( \mathfrak{a}\right) = {C}_{0} \) in this case,... | Yes |
Lemma 10.66. Let \( E \) be a standard subset of \( \mathcal{M} \) with \( k \) parameters, and let \( \gamma \) be its associated translation. Then the number of monomials \( {X}^{\alpha } \) with \( \left| \gamma \right| \leq s \) such that \( \alpha \) is in \( E \) is equal to the binomial coefficient \[ \left( \be... | Proof. Let \( \left\langle {{e}_{{j}_{1}},\ldots ,{e}_{{j}_{k}}}\right\rangle \) be the associated vector subspace for \( E \) . The associated translation \( \gamma \) is assumed to have \( {\gamma }_{i} = 0 \) for \( i \) in \( \left\{ {{j}_{1},\ldots ,{j}_{k}}\right\} \) . We are to count monomials \( {X}^{\alpha } ... | Yes |
Proposition 10.69. A polynomial \( P\left( s\right) \) in one variable of degree \( d \) takes integer values for \( s \) sufficiently large and positive if and only if it is an integer linear combination of the polynomials \( s \mapsto \left( \begin{array}{l} s \\ j \end{array}\right) \) for \( 0 \leq j \leq d \) . | Proof. The sufficiency is immediate because \( \left( \begin{array}{l} s \\ j \end{array}\right) \) is an integer for each \( j \) and \( s \) . For necessity, suppose that \( P\left( s\right) \) is integer-valued and has degree \( d \) . Since \( s \mapsto \left( \begin{array}{l} s \\ j \end{array}\right) \) is intege... | Yes |
Corollary 10.70. If \( \mathfrak{a} \) is a monomial ideal in \( \mathbb{k}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \) such that \( V\left( \mathfrak{a}\right) \) has geometric dimension \( d \), then the affine Hilbert polynomial \( {H}_{a}\left( {s,\mathfrak{a}}\right) \) of \( \mathfrak{a} \) is of the fo... | Proof. This follows by combining Theorem 10.68 and Proposition 10.69. | No |
Corollary 10.72. If \( \mathfrak{a} \) is an ideal in \( \mathbb{k}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \), then for all \( s \) sufficiently large, the affine Hilbert function \( {\mathcal{H}}_{a}\left( {s,\mathfrak{a}}\right) \) of \( \mathfrak{a} \) equals a polynomial in \( s \) of the form \( \matho... | Proof. Theorem 10.71 says that \( {\mathcal{H}}_{a}\left( {s,\mathfrak{a}}\right) = {\mathcal{H}}_{a}\left( {s,\operatorname{LT}\left( \mathfrak{a}\right) }\right) \) . Consequently the result follows immediately by applying Corollary 10.70 to LT(a). | Yes |
Corollary 10.73. If a graded monomial ordering is imposed on \( \mathbb{k}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \) and if \( \mathfrak{a} \) is any ideal in \( \mathbb{k}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \), then the affine Hilbert polynomials of \( \mathfrak{a} \) and \( \mathrm{{LT}}\l... | Proof. This is immediate from Theorem 10.71 and the definition of the affine Hilbert polynomial given in the remarks with Corollary 10.72. | No |
Corollary 10.74. If \( \\mathfrak{a} \) and \( \\mathfrak{b} \) are proper ideals of \( \\mathbb{k}\\left\\lbrack {{X}_{1},\\ldots ,{X}_{n}}\\right\\rbrack \) such that \( \\mathfrak{a} \\subseteq \\mathfrak{b} \), then \( \\deg {H}_{a}\\left( {s,\\mathfrak{a}}\\right) \\geq \\deg {H}_{a}\\left( {s,\\mathfrak{b}}\\righ... | Proof. Introduce a graded monomial ordering. The inclusion \( \\mathfrak{a} \\subseteq \\mathfrak{b} \) implies that \( \\operatorname{LT}\\left( \\mathfrak{a}\\right) \\subseteq \\operatorname{LT}\\left( \\mathfrak{b}\\right) \). Therefore \( \\mathcal{C}\\left( {\\operatorname{LT}\\left( \\mathfrak{a}\\right) }\\righ... | Yes |
Proposition 10.76. If \( \mathfrak{a} \) is a proper ideal in \( \mathbb{k}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \), then the degrees of the affine Hilbert polynomials \( {H}_{a}\left( {s,\mathfrak{a}}\right) \) and \( {H}_{a}\left( {s,\sqrt{\mathfrak{a}}}\right) \) are equal. | Proof. Fix a graded monomial ordering. We begin by proving that\n\n\[ \operatorname{LT}\left( \mathfrak{a}\right) \subseteq \operatorname{LT}\left( \sqrt{\mathfrak{a}}\right) \subseteq \sqrt{\operatorname{LT}\left( \mathfrak{a}\right) }.\]\n\n\( \left( *\right) \)\n\nThe left-hand inclusion is immediate because \( \mat... | Yes |
Corollary 10.77. If \( \mathfrak{a} \) and \( \mathfrak{b} \) are proper ideals in \( \mathbb{k}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \) with \( V\left( \mathfrak{a}\right) \subseteq \) \( V\left( \mathfrak{b}\right) \), then \( \deg {H}_{a}\left( {s,\mathfrak{a}}\right) \leq \deg {H}_{a}\left( {s,\mathfr... | Proof. Application of \( I\left( \cdot \right) \) to the inclusion \( V\left( \mathfrak{a}\right) \subseteq V\left( \mathfrak{b}\right) \) gives \( \sqrt{\mathfrak{a}} = \) \( I\left( {V\left( \mathfrak{a}\right) }\right) \supseteq I\left( {V\left( \mathfrak{b}\right) }\right) = \sqrt{\mathfrak{b}} \) . Then Corollary ... | Yes |
Corollary 10.80. If \( \mathfrak{a} \) is any ideal in \( \mathbb{k}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \), then the geometric dimension of the affine algebraic set \( V\left( \mathfrak{a}\right) \) equals the degree of the affine Hilbert polynomial \( {H}_{a}\left( {s,\mathfrak{a}}\right) \) . | Proof. Write \( V\left( \mathfrak{a}\right) = \mathop{\bigcup }\limits_{{j = 1}}^{k}{V}_{j} \) as a finite union of affine varieties \( {V}_{j} \), and define \( {\mathfrak{p}}_{j} = I\left( {V}_{j}\right) \) . Since \( {V}_{j} \) is irreducible, \( {\mathfrak{p}}_{j} \) is prime. Moreover, \( {V}_{j} = V\left( {I\left... | Yes |
Corollary 10.82. If \( \mathfrak{a} \) is a homogeneous ideal in \( \mathbb{k}\left\lbrack {{X}_{0},\ldots ,{X}_{n}}\right\rbrack \) and if the corresponding projective algebraic set \( V\left( \mathfrak{a}\right) \) is nonempty, then \( \dim V\left( \mathfrak{a}\right) \) equals the degree of the Hilbert polynomial \(... | Proof. This is immediate from Proposition 10.81 because \( \dim C\left( {V\left( \mathfrak{a}\right) }\right) = \) \( \dim {H}_{a}\left( {s,\mathfrak{a}}\right) \) and because \( \deg H\left( {s,\mathfrak{a}}\right) = \deg {H}_{a}\left( {s,\mathfrak{a}}\right) - 1 \) . | Yes |
Corollary 10.84. If \( \mathfrak{a} \) is any homogeneous ideal in \( \mathbb{k}\left\lbrack {{X}_{0},\ldots ,{X}_{n}}\right\rbrack \) and if \( {F}_{1},\ldots ,{F}_{r} \) are homogeneous polynomials, then\n\n\[ \dim V\left( \mathfrak{a}\right) \geq \dim V\left( {\mathfrak{a} + \left( {{F}_{1},\ldots ,{F}_{r}}\right) }... | Proof. We use Theorem 10.83 inductively, first applying it to the ideal \( \mathfrak{a} \) with \( F = {F}_{1} \), then applying it to the ideal \( \mathfrak{a} + \left( {F}_{1}\right) \) with \( F = {F}_{2} \), and so on. This proves the first conclusion, and the second conclusion follows because of the convention tha... | Yes |
Over an algebraically closed field any system of homogeneous polynomial equations with more variables than equations has a nonzero solution. | Let there be \( r \) equations and \( n + 1 \) variables with \( n + 1 > r \), the equations being \( {F}_{1} = 0,\ldots ,{F}_{r} = 0 \) . The zero locus for each equation is a subset of \( {\mathbb{P}}^{n} \) . Applying Corollary 10.84 with \( \mathfrak{a} = 0 \) shows that \( \dim V\left( {{F}_{1},\ldots ,{F}_{r}}\ri... | Yes |
Proposition 1.1 (division algorithm). If \( a \) and \( b \) are integers with \( b \neq 0 \), then there exist unique integers \( q \) and \( r \) such that \( a = {bq} + r \) and \( 0 \leq r < \left| b\right| \) . | Proof. Possibly replacing \( q \) by \( - q \), we may assume that \( b > 0 \) . The integers \( n \) with \( {bn} \leq a \) are bounded above by \( \left| a\right| \), and there exists such an \( n \), namely \( n = - \left| a\right| \) . Therefore there is a largest such integer, say \( n = q \) . Set \( r = \) \( a ... | Yes |
Proposition 1.2. Let \( a \) and \( b \) be integers with \( b \neq 0 \), and let \( d = \operatorname{GCD}\left( {a, b}\right) \). Then\n\n(a) the number \( {r}_{n} \) in the Euclidean algorithm is exactly \( d \),\n\n(b) any divisor \( {d}^{\prime } \) of both \( a \) and \( b \) necessarily divides \( d \),\n\n(c) t... | Proof of Proposition 1.2. Put \( {r}_{0} = b \) and \( {r}_{-1} = a \), so that\n\n\[ {r}_{k - 2} = {r}_{k - 1}{q}_{k} + {r}_{k}\;\text{ for }1 \leq k \leq n.\]\n\n\( \left( *\right) \)\n\nThe argument proceeds in three steps.\n\nStep 1. We show that \( {r}_{n} \) is a divisor of both \( a \) and \( b \) . In fact, fro... | Yes |
Corollary 1.3. Within \( \mathbb{Z} \), if \( c \) is a nonzero integer that divides a product \( {mn} \) and if \( \operatorname{GCD}\left( {c, m}\right) = 1 \), then \( c \) divides \( n \) . | Proof. Proposition 1.2c produces integers \( x \) and \( y \) with \( {cx} + {my} = 1 \) . Multiplying by \( n \), we obtain \( {cnx} + {mny} = n \) . Since \( c \) divides \( {mn} \) and divides itself, \( c \) divides both terms on the left side. Therefore it divides the right side, which is \( n \) . | Yes |
Corollary 1.4. Within \( \mathbb{Z} \), if \( a \) and \( b \) are nonzero integers with \( \operatorname{GCD}\left( {a, b}\right) = 1 \) and if both of them divide the integer \( m \), then \( {ab} \) divides \( m \) . | Proof. Proposition 1.2c produces integers \( x \) and \( y \) with \( {ax} + {by} = 1 \) . Multiplying by \( m \), we obtain \( {amx} + {bmy} = m \), which we rewrite in integers as \( {ab}\left( {m/b}\right) x + {ab}\left( {m/a}\right) y = m \) . Since \( {ab} \) divides each term on the left side, it divides the righ... | Yes |
Lemma 1.6. Within \( \mathbb{Z} \), if \( p \) is a prime and \( p \) divides a product \( {ab} \), then \( p \) divides \( a \) or \( p \) divides \( b \) . | Proof. Suppose that \( p \) does not divide \( a \) . Since \( p \) is prime, \( \operatorname{GCD}\left( {a, p}\right) = 1 \) . Taking \( m = a, n = b \), and \( c = p \) in Corollary 1.3, we see that \( p \) divides \( b \) . | Yes |
Corollary 1.7. If \( n = {p}_{1}^{{k}_{1}}\cdots {p}_{r}^{{k}_{r}} \) is a prime factorization of a positive integer \( n \), then the positive divisors \( d \) of \( n \) are exactly all products \( d = {p}_{1}^{{l}_{1}}\cdots {p}_{r}^{{l}_{r}} \) with \( 0 \leq {l}_{j} \leq {k}_{j} \) for all \( j \) . | Proof. Certainly any such product divides \( n \) . Conversely if \( d \) divides \( n \), write \( n = {dx} \) for some positive integer \( x \) . Apply Theorem 1.5 to \( d \) and to \( x \), form the resulting prime factorizations, and multiply them together. Then we see from the uniqueness for the prime factorizatio... | Yes |
Corollary 1.8. If two positive integers \( a \) and \( b \) have expansions as products of powers of \( r \) distinct primes given by \( a = {p}_{1}^{{k}_{1}}\cdots {p}_{r}^{{k}_{r}} \) and \( b = {p}_{1}^{{l}_{1}}\cdots {p}_{r}^{{l}_{r}} \), then\n\n\[ \operatorname{GCD}\left( {a, b}\right) = {p}_{1}^{\min \left( {{k}... | Proof. Let \( {d}^{\prime } \) be the right side of the displayed equation. It is plain that \( {d}^{\prime } \) is positive and that \( {d}^{\prime } \) divides \( a \) and \( b \) . On the other hand, two applications of Corollary 1.7 show that the greatest common divisor of \( a \) and \( b \) is a number \( d \) of... | Yes |
Corollary 1.9 (Chinese Remainder Theorem). Let \( a \) and \( b \) be positive relatively prime integers. To each pair \( \left( {r, s}\right) \) of integers with \( 0 \leq r < a \) and \( 0 \leq s < b \) corresponds a unique integer \( n \) such that \( 0 \leq n < {ab}, a \) divides \( n - r \), and \( b \) divides \(... | Proof. Let us see that \( n \) exists as asserted. Since \( a \) and \( b \) are relatively prime, Proposition 1.2c produces integers \( {x}^{\prime } \) and \( {y}^{\prime } \) such that \( a{x}^{\prime } - b{y}^{\prime } = 1 \) . Multiplying by \( s - r \), we obtain \( {ax} - {by} = s - r \) for suitable integers \(... | Yes |
Corollary 1.10. Let \( N > 1 \) be an integer, and let \( N = {p}_{1}^{{k}_{1}}\cdots {p}_{r}^{{k}_{r}} \) be a prime factorization of \( N \) . Then\n\n\[ \varphi \left( N\right) = \mathop{\prod }\limits_{{j = 1}}^{r}{p}_{j}^{{k}_{j} - 1}\left( {{p}_{j} - 1}\right) \] | Proof. For positive integers \( a \) and \( b \), let us check that\n\n\[ \varphi \left( {ab}\right) = \varphi \left( a\right) \varphi \left( b\right) \;\text{ if }\;\operatorname{GCD}\left( {a, b}\right) = 1. \]\n\n\( \left( *\right) \)\n\nIn view of Corollary 1.9, it is enough to prove that the mapping \( \left( {r, ... | Yes |
Corollary 1.11. Let \( {a}_{1},\ldots ,{a}_{t} \) be positive integers, and let \( d \) be their greatest common divisor. Then\n\n(a) if for each \( j \) with \( 1 \leq j \leq t,{a}_{j} = {p}_{1}^{{k}_{1, j}}\cdots {p}_{r}^{{k}_{r, j}} \) is an expansion of \( {a}_{j} \) as a product of powers of \( r \) distinct prime... | Proof. Part (a) is proved in the same way as Corollary 1.8 except that Corollary 1.7 is to be applied \( r \) times rather than just twice. Further application of Corollary 1.7 shows that any positive divisor \( {d}^{\prime } \) of \( {a}_{1},\ldots ,{a}_{t} \) is of the form \( {d}^{\prime } = {p}_{1}^{{m}_{1}}\cdots ... | Yes |
Proposition 1.12 (division algorithm). If \( A \) and \( B \) are polynomials in \( \mathbb{F}\left\lbrack X\right\rbrack \) and if \( B \) not the 0 polynomial, then there exist unique polynomials \( Q \) and \( R \) in \( \mathbb{F}\left\lbrack X\right\rbrack \) such that\n\n(a) \( A = {BQ} + R \) and\n\n(b) either \... | Proof of uniqueness. If \( A = {BQ} + R = B{Q}_{1} + {R}_{1} \), then \( B\left( {Q - {Q}_{1}}\right) = \) \( {R}_{1} - R \) . Without loss of generality, \( {R}_{1} - R \) is not the 0 polynomial since otherwise \( Q - {Q}_{1} = 0 \) also. Then\n\n\[ \deg B + \deg \left( {Q - {Q}_{1}}\right) = \deg \left( {{R}_{1} - R... | Yes |
Corollary 1.13 (Factor Theorem). If \( r \) is in \( \mathbb{F} \) and if \( P \) is a polynomial in \( \mathbb{F}\left\lbrack X\right\rbrack \), then \( X - r \) divides \( P \) if and only if \( P\left( r\right) = 0 \) . | Proof. If \( P = \left( {X - r}\right) Q \), then \( P\left( r\right) = \left( {r - r}\right) Q\left( r\right) = 0 \) . Conversely let \( P\left( r\right) = 0 \) . Taking \( B\left( X\right) = X - r \) in the division algorithm (Proposition 1.12), we obtain \( P = \left( {X - r}\right) Q + R \) with \( R = 0 \) or \( \... | Yes |
Corollary 1.14. If \( P \) is a nonzero polynomial with coefficients in \( \mathbb{F} \) and if \( \deg P = n \), then \( P \) has at most \( n \) distinct roots. | Proof. Let \( {r}_{1},\ldots ,{r}_{n + 1} \) be distinct roots of \( P\left( X\right) \) . By the Factor Theorem (Corollary 1.13), \( X - {r}_{1} \) is a factor of \( P\left( X\right) \) . We prove inductively on \( k \) that the product \( \left( {X - {r}_{1}}\right) \left( {X - {r}_{2}}\right) \cdots \left( {X - {r}_... | Yes |
Proposition 1.15. Let \( A \) and \( B \) be polynomials in \( \mathbb{F}\left\lbrack X\right\rbrack \) with \( B \neq 0 \), and let \( {R}_{1},\ldots ,{R}_{n} \) be the remainders generated by the Euclidean algorithm when applied to \( A \) and \( B \) . Then\n\n(a) \( {R}_{n} \) is a greatest common divisor of \( A \... | Proof. Conclusions (a) and (b) are proved in the same way that parts (a) and (b) of Proposition 1.2 are proved, and conclusion (d) is proved with \( D = {R}_{n} \) in the same way that Proposition \( {1.2}\mathrm{c} \) is proved.\n\nIf \( D \) is a greatest common divisor of \( A \) and \( B \), it follows from (a) and... | Yes |
Lemma 1.16. If \( A \) and \( B \) are nonzero polynomials with coefficients in \( \mathbb{F} \) and if \( P \) is a prime polynomial such that \( P \) divides \( {AB} \), then \( P \) divides \( A \) or \( P \) divides \( B \) . | Proof. If \( P \) does not divide \( A \), then 1 is a greatest common divisor of \( A \) and \( P \) , and Proposition 1.15d produces polynomials \( S \) and \( T \) such that \( {AS} + {PT} = 1 \) . Multiplication by \( B \) gives \( {ABS} + {PTB} = B \) . Then \( P \) divides \( {ABS} \) because it divides \( {AB} \... | Yes |
Theorem 1.17 (unique factorization). Every member of \( \mathbb{F}\left\lbrack X\right\rbrack \) of degree \( \geq 1 \) is a product of primes. This factorization is unique up to order and up to multiplication of each prime factor by a unit, i.e., by a nonzero scalar. | Proof. The existence follows in the same way as the existence in Theorem 1.5 ; induction on the integers is to be replaced by induction on the degree. The uniqueness follows from Lemma 1.16 in the same way that the uniqueness in Theorem 1.5 follows from Lemma 1.6. | No |
Corollary 1.19. Let \( P \) be a nonzero polynomial of degree \( n \) in \( \mathbb{C}\left\lbrack X\right\rbrack \) , and let \( {r}_{1},\ldots ,{r}_{k} \) be the distinct roots. Then there exist unique integers \( {m}_{j} > 0 \) for \( 1 \leq j \leq k \) such that \( P\left( X\right) \) is a scalar multiple of \( \ma... | Proof. We may assume that \( \deg P > 0 \) . We apply unique factorization (Theorem 1.17) to \( P\left( X\right) \) . It follows from the Fundamental Theorem of Algebra (Theorem 1.18) and the Factor Theorem (Corollary 1.13) that each prime polynomial with coefficients in \( \mathbb{C} \) has degree 1 . Thus the unique ... | Yes |
Proposition 1.20. Any polynomial in \( \mathbb{R}\left\lbrack X\right\rbrack \) with odd degree has at least one root. | Proof. Without loss of generality, we may take the leading coefficient to be 1. Thus let the polynomial be \( P\left( X\right) = {X}^{{2n} + 1} + {a}_{2n}{X}^{2n} + \cdots + {a}_{1}X + {a}_{0} = \) \( {X}^{{2n} + 1} + R\left( X\right) \) . For \( \left| r\right| \geq 1 \), the polynomial \( R \) satisfies \( \left| {R\... | Yes |
Proposition 1.21. Any permutation \( \sigma \) of \( \{ 1,2,\ldots, n\} \) is a product of disjoint cycles. The individual cycles in the decomposition are unique in the sense of being determined by \( \sigma \) . | Proof. Let us prove existence. Working with \( \{ 1,2,\ldots, n\} \), we show that any \( \sigma \) is the disjoint product of cycles in such a way that no cycle moves an element \( j \) unless \( \sigma \) moves \( j \) . We do so for all \( \sigma \) simultaneously by induction downward on the number of elements fixe... | Yes |
Any \( k \) -cycle \( \sigma \) permuting \( \{ 1,2,\ldots, n\} \) is a product of \( k - 1 \) transpositions if \( k > 1 \) . Therefore any permutation \( \sigma \) of \( \{ 1,2,\ldots, n\} \) is a product of transpositions. | For the first statement, we observe that \( \left( \begin{array}{lllll} {c}_{1} & {c}_{2} & \cdots & {c}_{k - 1} & {c}_{k} \end{array}\right) = \) \( \left( \begin{array}{ll} {c}_{1} & {c}_{k} \end{array}\right) \left( \begin{array}{ll} {c}_{1} & {c}_{k - 1} \end{array}\right) \cdots \left( \begin{array}{ll} {c}_{1} & ... | Yes |
Proposition 1.24. The signs of permutations of \( \{ 1,2,\ldots, n\} \) have the following properties:\n\n(a) \( \operatorname{sgn}1 = + 1 \) ,\n\n(b) \( \operatorname{sgn}\sigma = {\left( -1\right) }^{k} \) if \( \sigma \) can be written as the product of \( k \) transpositions,\n\n(c) \( \operatorname{sgn}\left( {\si... | Proof. Conclusion (a) is immediate from the definition. For (b), let \( \sigma = \) \( {\tau }_{1}\cdots {\tau }_{k} \) with each \( {\tau }_{j} \) equal to a transposition. We apply Lemma 1.23 recursively, using (a) at the end:\n\n\[ \operatorname{sgn}\left( {{\tau }_{1}\cdots {\tau }_{k}}\right) = \left( {-1}\right) ... | Yes |
Proposition 1.26. In the solution process for a system of \( k \) linear equations in \( n \) variables with the vertical line in place,\n\n(a) the sum of the number of corner variables and the number of independent variables is \( n \) ,\n\n(b) the number of corner variables equals the number of nonzero rows on the le... | Proof. Conclusions (a), (b), and (c) follow immediately by inspection of the solution method. For (d), we observe that no contradictory equation can arise when the right sides are 0 and, in addition, that there must be at least one independent variable by (a) since (b) shows that the number of corner variables is \( \l... | Yes |
Proposition 1.27. For an array with \( k \) rows and \( n \) columns in reduced row-echelon form,\n\n(a) the sum of the number of corner variables and the number of independent variables is \( n \) ,\n\n(b) the number of corner variables equals the number of nonzero rows and hence is \( \leq k \) ,\n\n(c) when \( k = n... | Proof. Conclusions (a) and (b) are immediate by inspection. In (c), failure of the reduced row-echelon form to be as indicated forces there to be some noncorner variable, so that the number of corner variables is \( < n \) . By (b), the number of nonzero rows is \( < n \), and hence there is a row of 0 ’s. | Yes |
Proposition 1.28. Matrix multiplication has the properties that\n\n(a) it is associative in the sense that \( \\left( {AB}\\right) C = A\\left( {BC}\\right) \), provided that the sizes match correctly, i.e., \( A \) is in \( {M}_{km}\\left( \\mathbb{F}\\right), B \) is in \( {M}_{mn}\\left( \\mathbb{F}\\right) \), and ... | PROOF. For (a), we have\n\n\[ \n{\\left( \\left( AB\\right) C\\right) }_{ij} = \\mathop{\\sum }\\limits_{{t = 1}}^{n}{\\left( AB\\right) }_{it}{C}_{tj} = \\mathop{\\sum }\\limits_{{t = 1}}^{n}\\;\\mathop{\\sum }\\limits_{{s = 1}}^{m}{A}_{is}{B}_{st}{C}_{tj} \n\]\n\nand\n\nand these are equal. For the first identity in ... | Yes |
Proposition 1.29. Each elementary row operation is given by left multiplication by an invertible matrix. The inverse matrix is the matrix of another elementary row operation. | Proof. For the interchange of rows \( i \) and \( j \), the part of the elementary matrix in the rows and columns with \( i \) or \( j \) as index is\n\n\[ \begin{matrix} i & j \\ i & \left( \begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right) \end{matrix} \]\n\nand otherwise the matrix is the identity. This matrix is i... | Yes |
Theorem 1.30. The following conditions on an \( n \) -by- \( n \) square matrix \( A \) are equivalent:\n\n(a) the reduced row-echelon form of \( A \) is the identity,\n\n(b) \( A \) is the product of elementary matrices,\n\n(c) \( A \) has an inverse,\n\n(d) the system of equations \( {AX} = 0 \) with \( X = \left( \b... | Proof. If (a) holds, choose a sequence of elementary row operations that reduce \( A \) to the identity, and let \( {E}_{1},\ldots ,{E}_{r} \) be the corresponding elementary matrices given by Proposition 1.29. Then we have \( {E}_{r}\cdots {E}_{1}A = I \), and hence \( A = {E}_{1}^{-1}\cdots {E}_{r}^{-1} \) . The prop... | Yes |
Corollary 1.31. If the solution procedure for finding the inverse of a square matrix \( A \) leads from \( \left( {A \mid I}\right) \) to \( \left( {I \mid X}\right) \), then \( A \) is invertible and its inverse is \( X \) . Conversely if the solution procedure leads to \( \left( {R \mid Y}\right) \) and \( R \) has a... | Proof. We apply the equivalence of (a) and (c) in Theorem 1.30 to settle the existence or nonexistence of \( {A}^{-1} \) . In the case that \( {A}^{-1} \) exists, we know that the solution procedure has to yield the inverse. | No |
Corollary 1.32. Let \( A \) be a square matrix. If \( B \) is a square matrix such that \( {BA} = I \), then \( A \) is invertible and \( B \) is its inverse. If \( C \) is a square matrix such that \( {AC} = I \), then \( A \) is invertible with inverse \( C \) . | Proof. Suppose \( {BA} = I \) . Let \( X \) be a column vector with \( {AX} = 0 \) . Then \( X = {IX} = \left( {BA}\right) X = B\left( {AX}\right) = {B0} = 0 \) . Since (d) implies (c) in Theorem 1.30, \( A \) is invertible. Suppose \( {AC} = I \) . Applying the result of the previous paragraph to \( C \), we conclude ... | Yes |
Proposition 2.1. Let \( V \) be a vector space over \( \mathbb{F} \). (a) If \( \left\{ {v}_{\alpha }\right\} \) is a linearly independent subset of \( V \) that is maximal with respect to the property of being linearly independent (i.e., has the property of being strictly contained in no linearly independent set), the... | Proof. For (a), let \( v \) be given. We are to show that \( v \) is in the span of \( \left\{ {v}_{\alpha }\right\} \). Without loss of generality, we may assume that \( v \) is not in the set \( \left\{ {v}_{\alpha }\right\} \) itself. By the assumed maximality, \( \left\{ {v}_{\alpha }\right\} \cup \{ v\} \) is not ... | Yes |
Proposition 2.2. Let \( V \) be a vector space over \( \mathbb{F} \) . If \( V \) has a finite spanning set \( \left\{ {{v}_{1},\ldots ,{v}_{m}}\right\} \), then any linearly independent set in \( V \) has \( \leq m \) elements. | Proof. It is enough to show that no subset of \( m + 1 \) vectors can be linearly independent. Arguing by contradiction, suppose that \( \left\{ {{u}_{1},\ldots ,{u}_{n}}\right\} \) is a linearly independent set with \( n = m + 1 \) . Write\n\n\[ \n{u}_{1} = {c}_{11}{v}_{1} + {c}_{21}{v}_{2} + \cdots + {c}_{m1}{v}_{m} ... | Yes |
Corollary 2.3. If the vector space \( V \) has a finite spanning set \( \left\{ {{v}_{1},\ldots ,{v}_{m}}\right\} \) , then\n\n(a) \( \left\{ {{v}_{1},\ldots ,{v}_{m}}\right\} \) has a subset that is a basis,\n\n(b) any linearly independent set in \( V \) can be extended to a basis,\n\n(c) \( V \) has a basis,\n\n(d) a... | Proof. By discarding elements of the set \( \left\{ {{v}_{1},\ldots ,{v}_{m}}\right\} \) one at a time if necessary and by applying Proposition 2.1b, we obtain (a). For (b), we see from Proposition 2.2 that the given linearly independent set has \( \leq m \) elements. If we adjoin elements to it one at a time so as to ... | Yes |
Corollary 2.4. If \( V \) is a finite-dimensional vector space with \( \dim V = n \), then any spanning set of \( n \) elements is a basis of \( V \), and any linearly independent set of \( n \) elements is a basis of \( V \) . Consequently any \( n \) -dimensional vector subspace \( U \) of \( V \) coincides with \( V... | Proof. These conclusions are immediate from parts (a) and (b) of Corollary 2.3 if we take part (d) into account. | No |
Corollary 2.5. If \( V \) is a finite-dimensional vector space and \( U \) is a vector subspace of \( V \), then \( U \) is finite-dimensional, and \( \dim U \leq \dim V \) . | Proof. Let \( \left\{ {{v}_{1},\ldots ,{v}_{m}}\right\} \) be a basis of \( V \) . According to Proposition 2.2, any linearly independent set in \( U \) has \( \leq m \) elements, being linearly independent in \( V \) . We can thus choose a maximal linearly independent subset of \( U \) with \( \leq m \) elements, and ... | Yes |
Proposition 2.7. If \( A \) is in \( {M}_{kn}\left( \mathbb{F}\right) \), then each elementary row operation on \( A \) preserves the row space of \( A \) . | Proof. Let the rows of \( A \) be \( {r}_{1},\ldots ,{r}_{k} \) . Their span is unchanged if we interchange two of them or multiply one of them by a nonzero scalar. If we replace the row \( {r}_{i} \) by \( {r}_{i} + c{r}_{j} \) with \( j \neq i \), then the span is unchanged since\n\n\[ \n{a}_{i}{r}_{i} + {a}_{j}{r}_{... | Yes |
Theorem 2.8. If \( A \) in \( {M}_{kn}\left( \mathbb{F}\right) \) has reduced row-echelon form \( R \), then\n\n\[ \dim \left( {\operatorname{row}\operatorname{space}\left( A\right) }\right) = \dim \left( {\operatorname{row}\operatorname{space}\left( R\right) }\right) \]\n\n\[ = \# \left( {\text{ nonzero rows of }R}\ri... | Proof. The first equality in the first conclusion is immediate from Proposition 2.7 , and the last equality of that conclusion is known from the method of row reduction. To see the middle inequality, we need to see that the nonzero rows of \( R \) are linearly independent. Let these rows be \( {r}_{1},\ldots ,{r}_{t} \... | Yes |
Corollary 2.10. If \( A \) is in \( {M}_{kn}\left( \mathbb{F}\right) \), then\n\n\[ \n\dim \left( {\operatorname{row}\operatorname{space}\left( A\right) }\right) = \dim \left( {\operatorname{column}\operatorname{space}\left( A\right) }\right) .\n\] | Proof. This follows by comparing Theorem 2.6 and Corollary 2.9. | No |
Proposition 2.11. If \( L : {\mathbb{F}}^{n} \rightarrow {\mathbb{F}}^{k} \) is a linear map, then there exists a unique \( k \) -by- \( n \) matrix \( A \) such that \( L\left( v\right) = {Av} \) for all \( v \) in \( {\mathbb{F}}^{n} \) . | REMARK. The proof will show how to obtain the matrix \( A \) .\n\nProof. For \( 1 \leq j \leq n \), let \( {e}_{j} \) be the \( {j}^{\text{th }} \) standard basis vector of \( {\mathbb{F}}^{n} \), having 1 in its \( {j}^{\text{th }} \) entry and 0 ’s elsewhere, and let the \( {j}^{\text{th }} \) column of \( A \) be th... | Yes |
Proposition 2.12. Let \( L : {\mathbb{F}}^{n} \rightarrow {\mathbb{F}}^{m} \) be the linear map corresponding to an \( m \) -by- \( n \) matrix \( A \), and let \( M : {\mathbb{F}}^{m} \rightarrow {\mathbb{F}}^{k} \) be the linear map corresponding to a \( k \) -by- \( m \) matrix \( B \) . Then the composite function ... | Proof. The function \( M \circ L \) satisfies \( \left( {M \circ L}\right) \left( {u + v}\right) = M\left( {L\left( {u + v}\right) }\right) = \) \( M\left( {{Lu} + {Lv}}\right) = M\left( {Lu}\right) + M\left( {Lv}\right) = \left( {M \circ L}\right) \left( u\right) + \left( {M \circ L}\right) \left( v\right) \), and sim... | Yes |
Proposition 2.13. Let \( U \) and \( V \) be vector spaces over \( \mathbb{F} \), and let \( \Gamma \) be a basis of \( U \) . Then to each function \( \ell : \Gamma \rightarrow V \) corresponds one and only one linear map \( L : U \rightarrow V \) whose restriction to \( \Gamma \) has \( {\left. L\right| }_{\Gamma } =... | Proof. Suppose that \( \ell : \Gamma \rightarrow V \) is given. Since \( \Gamma \) is a basis of \( U \), each element of \( U \) has a unique expansion as a finite linear combination of members of \( \Gamma \) . Say that \( u = {c}_{{\alpha }_{1}}{u}_{{\alpha }_{1}} + \cdots + {c}_{{\alpha }_{r}}{u}_{{\alpha }_{r}} \)... | Yes |
Theorem 2.14. If \( L : U \rightarrow V \) is a linear map between finite-dimensional vector spaces over \( \mathbb{F} \) and if \( \Gamma \) and \( \Delta \) are ordered bases of \( U \) and \( V \), respectively, then \[ \left( \begin{matrix} L\left( u\right) \\ \Delta \end{matrix}\right) = \left( \begin{matrix} L \\... | Proof. The two sides of the identity in question are linear in \( u \), and Proposition 2.13 shows that it is enough to prove the identity for the members \( u \) of some ordered basis of \( U \) . We choose \( \Gamma \) as this ordered basis. For the basis vector \( u \) equal to the \( {j}^{\text{th }} \) member \( {... | Yes |
Theorem 2.16. Let \( L : U \rightarrow V \) and \( M : V \rightarrow W \) be linear maps between finite-dimensional vector spaces, and let \( \Gamma ,\Delta \), and \( \Omega \) be ordered bases of \( U \) , \( V \), and \( W \) . Then the composition \( {ML} \) is linear, and the corresponding matrix is given by\n\n\[... | Proof. If \( u \) is in \( U \), three applications of Theorem 2.14 and one application of associativity of matrix multiplication give\n\n\[ \left( \begin{matrix} {ML} \\ {\Omega \Gamma } \end{matrix}\right) \left( \begin{matrix} u \\ \Gamma \end{matrix}\right) = \left( \begin{matrix} {ML}\left( u\right) \\ \Omega \end... | Yes |
Proposition 2.18. Two finite-dimensional vector spaces over \( \mathbb{F} \) are isomorphic if and only if they have the same dimension. | Proof. If a vector space \( U \) is isomorphic to a vector space \( V \), then the isomorphism carries any basis of \( U \) to a basis of \( V \), and hence \( U \) and \( V \) have the same dimension. Conversely if they have the same dimension, let \( \left( {{u}_{1},\ldots ,{u}_{n}}\right) \) be an ordered basis of \... | Yes |
Proposition 2.20. Let \( V \) be a finite-dimensional vector space, and let \( U \) be a vector subspace of \( V \) . Then\n\n(a) \( \dim U + \dim \operatorname{Ann}\left( U\right) = \dim V \) ,\n\n(b) every linear functional on \( U \) extends to a linear functional on \( V \) ,\n\n(c) whenever \( {v}_{0} \) is a memb... | Proof. We retain the notation above, writing \( \left\{ {{v}_{1},\ldots ,{v}_{r}}\right\} \) for a basis of \( U \) , \( {v}_{r + 1},\ldots ,{v}_{n} \) for vectors that are adjoined to form a basis of \( V \), and \( \left\{ {{v}_{1}^{\prime },\ldots ,{v}_{n}^{\prime }}\right\} \) for the dual basis of \( {V}^{\prime }... | Yes |
Proposition 2.21. Let \( L : U \rightarrow V \) be a linear map between finite-dimensional vector spaces, let \( {L}^{t} : {V}^{\prime } \rightarrow {U}^{\prime } \) be its contragredient, let \( \Gamma \) and \( \Delta \) be respective ordered bases of \( U \) and \( V \), and let \( {\Gamma }^{\prime } \) and \( {\De... | Proof. Let \( \Gamma = \left( {{u}_{1},\ldots ,{u}_{n}}\right) ,\Delta = \left( {{v}_{1},\ldots ,{v}_{k}}\right) ,{\Gamma }^{\prime } = \left( {{u}_{1}^{\prime },\ldots ,{u}_{n}^{\prime }}\right) \), and \( {\Delta }^{\prime } = \left( {{v}_{1}^{\prime },\ldots ,{v}_{k}^{\prime }}\right) \) . Write \( B \) and \( A \) ... | Yes |
Proposition 2.22. If \( V \) is any finite-dimensional vector space over \( \mathbb{F} \), then the canonical map \( \iota : V \rightarrow {V}^{\prime \prime } \) is one-one onto. | Proof. We saw in Section 3 that a linear map \( \iota \) is one-one if and only if \( \ker \iota = 0 \) . Thus suppose \( \iota \left( v\right) = 0 \) . Then \( 0 = \iota \left( v\right) \left( {v}^{\prime }\right) = {v}^{\prime }\left( v\right) \) for all \( {v}^{\prime } \) . Arguing by contradiction, suppose \( v \n... | Yes |
Proposition 2.23. Let \( V \) be a vector space over \( \mathbb{F} \), and let \( U \) be a vector subspace. The relation defined by saying that \( {v}_{1} \sim {v}_{2} \) if \( {v}_{1} - {v}_{2} \) is in \( U \) is an equivalence relation, and the equivalence classes are all sets of the form \( v + U \) with \( v \in ... | Proof. The properties of an equivalence relation are established as follows:\n\n\( {v}_{1} \sim {v}_{1} \) because 0 is in \( U \),\n\n\( {v}_{1} \sim {v}_{2} \) implies \( {v}_{2} \sim {v}_{1} \) because \( U \) is closed under negatives,\n\n\( {v}_{1} \sim {v}_{2} \) and \( {v}_{2} \sim {v}_{3} \)\ntogether imply \( ... | Yes |
Corollary 2.24. If \( V \) is a vector space over \( \mathbb{F} \) and \( U \) is a vector subspace, then\n\n(a) \( \dim V = \dim U + \dim \left( {V/U}\right) \) ,\n\n(b) the subspace \( U \) is the kernel of some linear map defined on \( V \) . | Proof. Let \( q \) be the quotient map. The linear map \( q \) meets the conditions of (b). For (a), take a basis of \( U \) and extend to a basis of \( V \) . Then the images under \( q \) of the additional vectors form a basis of \( V/U \) . | Yes |
Proposition 2.25. Let \( L : V \rightarrow W \) be a linear map between vector spaces over \( \mathbb{F} \), let \( {U}_{0} = \ker L \), let \( U \) be a vector subspace of \( V \) contained in \( {U}_{0} \), and let \( q : V \rightarrow V/U \) be the quotient map. Then there exists a linear map \( \bar{L} : V/U \right... | Proof. The definition of \( \bar{L} \) has to be \( \bar{L}\left( {v + U}\right) = L\left( v\right) \) . This forces \( \bar{L}q = L \) , and \( \bar{L} \) will have to be linear. What needs proof is that \( \bar{L} \) is well defined. Thus suppose \( {v}_{1} \sim {v}_{2} \) . We are to prove that \( \bar{L}\left( {{v}... | Yes |
Corollary 2.26. Let \( L : V \rightarrow W \) be a linear map between vector spaces over \( \mathbb{F} \), and suppose that \( L \) is onto \( W \) and has kernel \( U \) . Then \( V/U \) is canonically isomorphic to \( W \) . | Proof. Take \( U = {U}_{0} \) in Proposition 2.25, and form \( \bar{L} : V/U \rightarrow W \) with \( L = \bar{L}q \) . The proposition shows that \( \bar{L} \) is onto \( W \) and has trivial kernel, i.e., the 0 element of \( V/U \) . Having trivial kernel, \( \bar{L} \) is one-one. | Yes |
Theorem 2.27 (First Isomorphism Theorem). Let \( L : V \rightarrow W \) be a linear map between vector spaces over \( \mathbb{F} \), and suppose that \( L \) is onto \( W \) and has kernel \( U \) . Then the map \( S \mapsto L\left( S\right) \) gives a one-one correspondence between\n\n(a) the vector subspaces \( S \) ... | Proof. The passage from (a) to (b) is by direct image under \( L \), and the passage from (b) to (a) will be by inverse image under \( {L}^{-1} \) . Certainly the direct image of a vector subspace as in (a) is a vector subspace as in (b). We are to show that the inverse image of a vector subspace as in (b) is a vector ... | Yes |
Theorem 2.28 (Second Isomorphism Theorem). Let \( M \) and \( N \) be vector subspaces of a vector space \( V \) over \( \mathbb{F} \). Then the map \( n + \left( {M \cap N}\right) \mapsto n + M \) is a well-defined canonical vector-space isomorphism\n\n\[ N/\left( {M \cap N}\right) \cong \left( {M + N}\right) /M. \] | Proof. The function \( L\left( {n + \left( {M \cap N}\right) }\right) = n + M \) is well defined since \( M \cap N \subseteq \) \( M \), and \( L \) is linear. The domain of \( L \) is \( \{ n + \left( {M \cap N}\right) \mid n \in N\} \), and the kernel is the subset of this where \( n \) lies in \( M \) as well as \( ... | Yes |
Corollary 2.29. Let \( M \) and \( N \) be finite-dimensional vector subspaces of a vector space \( V \) over \( \mathbb{F} \). Then\n\n\[ \dim \left( {M + N}\right) + \dim \left( {M \cap N}\right) = \dim M + \dim N. \] | Proof. Theorem 2.28 and two applications of Corollary 2.24a yield\n\n\[ \dim \left( {M + N}\right) - \dim M = \dim \left( {\left( {M + N}\right) /M}\right) \]\n\n\[ = \dim \left( {N/\left( {M \cap N}\right) }\right) = \dim N - \dim \left( {M \cap N}\right) ,\]\n\nand the result follows. | Yes |
Proposition 2.30. Let \( V \) be a vector space over \( \mathbb{F} \), and let \( {V}_{1} \) and \( {V}_{2} \) be vector subspaces of \( V \). Then the following conditions are equivalent:\n\n(a) every member \( v \) of \( V \) decomposes uniquely as \( v = {v}_{1} + {v}_{2} \) with \( {v}_{1} \in {V}_{1} \) and \( {v}... | Proof. If (a) holds, then the existence of the decomposition \( v = {v}_{1} + {v}_{2} \) shows that \( {V}_{1} + {V}_{2} = V \). If \( v \) is in \( {V}_{1} \cap {V}_{2} \), then \( 0 = v + \left( {-v}\right) \) is a decomposition of the kind in (a), and the uniqueness forces \( v = 0 \). Therefore \( {V}_{1} \cap {V}_... | Yes |
Proposition 2.31. Let \( V \) be a vector space over \( \mathbb{F} \), and let \( {V}_{1},\ldots ,{V}_{n} \) be vector subspaces of \( V \) . Then the following conditions are equivalent:\n\n(a) every member \( v \) of \( V \) decomposes uniquely as \( v = {v}_{1} + \cdots + {v}_{n} \) with \( {v}_{j} \in {V}_{j} \) fo... | Proposition 2.31 is proved in the same way as Proposition 2.30, and the expected analog of Remark 3 with that proposition is valid as well. | No |
Proposition 2.32. Let \( A \) be a nonempty set of vector spaces over \( \mathbb{F} \), and let \( {V}_{\alpha } \) be the vector space corresponding to the member \( \alpha \) of \( A \) . If \( \left( {V,\left\{ {p}_{\alpha }\right\} }\right) \) and \( \left( {{V}^{ * },\left\{ {p}_{\alpha }^{ * }\right\} }\right) \)... | Proof. In Figure 2.4 let \( U = {V}^{ * } \) and \( {L}_{\alpha } = {p}_{\alpha }^{ * } \) . If \( L : {V}^{ * } \rightarrow V \) is the linear map produced by the fact that \( V \) is a direct product, then we have \( {p}_{\alpha }L = {p}_{\alpha }^{ * } \) for all \( \alpha \) . Reversing the roles of \( V \) and \( ... | Yes |
Proposition 2.33. Let \( A \) be a nonempty set of vector spaces over \( \mathbb{F} \), and let \( {V}_{\alpha } \) be the vector space corresponding to the member \( \alpha \) of \( A \) . If \( \left( {V,\left\{ {i}_{\alpha }\right\} }\right) \) and \( \left( {{V}^{ * },\left\{ {i}_{\alpha }^{ * }\right\} }\right) \)... | Proof. In Figure 2.5 let \( W = {V}^{ * } \) and \( {M}_{\alpha } = {i}_{\alpha }^{ * } \) . If \( M : V \rightarrow {V}^{ * } \) is the linear map produced by the fact that \( V \) is a direct sum, then we have \( M{i}_{\alpha } = {i}_{\alpha }^{ * } \) for all \( \alpha \) . Reversing the roles of \( V \) and \( {V}^... | Yes |
For \( {M}_{1n}\left( \mathbb{F}\right) \), the vector space of alternating \( n \) -multilinear functionals has dimension 1 , and a nonzero such functional has nonzero value on \( \left( {{e}_{1}^{t},\ldots ,{e}_{n}^{t}}\right) \), where \( \left\{ {{e}_{1},\ldots ,{e}_{n}}\right\} \) is the standard basis of \( {\mat... | Proof of UNIQUENESS. Let \( f \) be an alternating \( n \) -multilinear functional, and let \( \left\{ {{u}_{1},\ldots ,{u}_{n}}\right\} \) be the basis of the space of row vectors defined by \( {u}_{i} = {e}_{i}^{t} \) . Since \( f \) is multilinear, \( f \) is determined by its values on all \( n \) -tuples \( \left(... | Yes |
Proposition 2.35. If \( A \) is an \( n \) -by- \( n \) square matrix, then \( \det {A}^{t} = \det A \) . | Proof. Corollary 2.9 says that the row space and the column space of \( A \) have the same dimension, and \( A \) is invertible if and only if the row space has dimension \( n \) . Thus \( A \) is invertible if and only if \( {A}^{t} \) is invertible, and Theorem 2.34c thus shows that \( \det A = 0 \) if and only if \(... | Yes |
Corollary 2.37 (Vandermonde matrix and determinant). If \( {r}_{1},\ldots ,{r}_{n} \) are scalars, then\n\n\[ \det \left( \begin{matrix} 1 & 1 & \cdots & 1 \\ {r}_{1} & {r}_{2} & \cdots & {r}_{n} \\ {r}_{1}^{2} & {r}_{2}^{2} & \cdots & {r}_{n}^{2} \\ \vdots & \vdots & \ddots & \vdots \\ {r}_{1}^{n - 1} & {r}_{2}^{n - 1... | Proof. We show that the determinant is\n\n\[ = \mathop{\prod }\limits_{{j > 1}}\left( {{r}_{j} - {r}_{1}}\right) \det \left( \begin{matrix} 1 & \cdots & 1 \\ {r}_{2} & \cdots & {r}_{n} \\ \vdots & \ddots & \vdots \\ {r}_{2}^{n - 2} & \cdots & {r}_{n}^{n - 2} \end{matrix}\right) \]\n\nand then the result follows by indu... | Yes |
Proposition 2.38 (Cramer’s rule). If \( A \) is an \( n \) -by- \( n \) matrix, then \( A{A}^{\text{adj }} = \) \( {A}^{\text{adj }}A = \left( {\det A}\right) I \), and thus \( \det A \neq 0 \) implies \( {A}^{-1} = {\left( \det A\right) }^{-1}{A}^{\text{adj }} \) . Consequently if \( \det A \neq 0 \), then the unique ... | Proof. The \( {\left( i, j\right) }^{\text{th }} \) entry of \( {A}^{\text{adj }}A \) is \[ {\left( {A}^{\mathrm{{adj}}}A\right) }_{ij} = \mathop{\sum }\limits_{{k = 1}}^{n}{A}_{ik}^{\mathrm{{adj}}}{A}_{kj} = \mathop{\sum }\limits_{{k = 1}}^{n}{\left( -1\right) }^{i + k}\left( {\det \widehat{{A}_{ki}}}\right) {A}_{kj}.... | Yes |
An \( n \) -by- \( n \) matrix \( A \) has an eigenvector with eigenvalue \( \lambda \) if and only if \( \det \left( {{\lambda I} - A}\right) = 0 \) . In this case the eigenspace for \( \lambda \) is the kernel of \( {\lambda I} - A \) . | We have \( {Av} = {\lambda v} \) if and only if \( \left( {{\lambda I} - A}\right) v = 0 \), if and only if \( v \) is in \( \ker \left( {{\lambda I} - A}\right) \) . This kernel is nonzero if and only if \( \det \left( {{\lambda I} - A}\right) = 0 \) . | Yes |
Corollary 2.40. An \( n \) -by- \( n \) matrix \( A \) has at most \( n \) eigenvalues. | Proof. Since \( \det \left( {{\lambda I} - A}\right) \) is a polynomial of degree \( n \), this follows from Proposition 2.39 and Corollary 1.14. | Yes |
Proposition 2.41. If \( A \) is an \( n \) -by- \( n \) matrix, then eigenvectors for distinct eigenvalues are linearly independent. | Proof. Let \( A{v}_{1} = {\lambda }_{1}{v}_{1},\ldots, A{v}_{k} = {\lambda }_{k}{v}_{k} \) with \( {\lambda }_{1},\ldots ,{\lambda }_{k} \) distinct, and suppose that\n\n\[ \n{c}_{1}{v}_{1} + \cdots + {c}_{k}{v}_{k} = 0.\n\]\n\nApplying \( A \) repeatedly gives\n\n\[ \n{c}_{1}{\lambda }_{1}{v}_{1} + \cdots + {c}_{k}{\l... | Yes |
If \( V \) is any vector space over \( \mathbb{F} \), then (b) any linearly independent set in \( V \) can be extended to a basis. | For (b), let \( E \) be the given linearly independent set, and let \( \mathcal{S} \) be the collection of all linearly independent subsets of \( V \) that contain \( E \) . Partially order \( \mathcal{S} \) by inclusion upward. The set \( \mathcal{S} \) is nonempty because \( E \) is in \( \mathcal{S} \) . Let \( \mat... | Yes |
Proposition 2.13. Let \( U \) and \( V \) be vector spaces over \( \mathbb{F} \), and let \( \Gamma \) be a basis of \( U \) . Then to each function \( \ell : \Gamma \rightarrow V \) corresponds one and only one linear map \( L : U \rightarrow V \) such that \( {\left. L\right| }_{\Gamma } = \ell \) . | In fact, the proof given in Section 3 is valid with no assumption about finite dimensionality. | Yes |
Proposition 2.18. Two vector spaces over \( \mathbb{F} \) are isomorphic if and only if they have the same cardinal-number dimension. | In fact, this result follows from Proposition 2.13 just as it did in the finite-dimensional case; the only changes that are needed in the argument in Section 3 are small adjustments of the notation. Of course, one must not overinterpret this result on the basis of the remark with Theorem 2.42: two vector spaces with di... | No |
Proposition 2.19. If \( V \) is a vector space and \( {V}^{\prime } \) is its dual, then \( \dim V \leq \) \( \dim {V}^{\prime } \) . (In the infinite-dimensional case we do not have equality.) | In fact, take a basis \( \left\{ {v}_{\alpha }\right\} \) of \( V \) . If for each \( \alpha \) we define \( {v}_{\alpha }^{\prime }\left( {v}_{\beta }\right) = {\delta }_{\alpha \beta } \) and use Proposition 2.13 to form the linear extension \( {v}_{\alpha }^{\prime } \), then the set \( \left\{ {v}_{\alpha }^{\prime... | No |
Proposition 2.20. Let \( V \) be a vector space, and let \( U \) be a vector subspace of \( V \). Then (b) every linear functional on \( U \) extends to a linear functional on \( V \). | To prove (b) without the finite dimensionality, let \( {u}^{\prime } \) be a given linear functional on \( U \), let \( \left\{ {u}_{\alpha }\right\} \) be a basis of \( U \), and let \( \left\{ {v}_{\beta }\right\} \) be a subset of \( V \) such that \( \left\{ {u}_{\alpha }\right\} \cup \left\{ {v}_{\beta }\right\} \... | Yes |
Proposition 2.22. If \( V \) is any vector space over \( \mathbb{F} \), then the canonical map \( \iota : V \rightarrow {V}^{\prime \prime } \) is one-one. The canonical map is not onto \( {V}^{\prime \prime } \) if \( V \) is infinite-dimensional. | The proof that it is one-one given in Section 4 is applicable in the infinite-dimensional case since we know from Theorem 2.42 that any linearly independent subset of \( V \) can be extended to a basis. For the second conclusion when \( V \) has a countably infinite basis, see Problem 31 at the end of the chapter. | No |
COROLLARY 2.24. If \( V \) is a vector space over \( \mathbb{F} \) and \( U \) is a vector subspace, then\n\n(a) \( \dim V = \dim U + \dim \left( {V/U}\right) \) ,\n\n(b) the subspace \( U \) is the kernel of some linear map defined on \( V \) . | The proof in Section 5 requires no changes: Let \( q \) be the quotient map. The linear map \( q \) meets the conditions of (b). For (a), take a basis of \( U \) and extend to a basis of \( V \) . Then the images under \( q \) of the additional vectors form a basis of \( V/U \) . | Yes |
Proposition 3.1 (Schwarz inequality). In any inner-product space \( V \) , \( \left| \left( {u, v}\right) \right| \leq \parallel u\parallel \parallel v\parallel \) for all \( u \) and \( v \) in \( V \) . | Possibly replacing \( u \) by \( {e}^{i\theta }u \) for some real \( \theta \), we may assume that \( \left( {u, v}\right) \) is real. In the case that \( \parallel v\parallel \neq 0 \), the law of cosines gives\n\n\[{\left| u - \parallel v{\parallel }^{-2}\left( u, v\right) v\right| }^{2} = \parallel u{\parallel }^{2}... | Yes |
Proposition 3.2. In any inner-product space \( V \), the norm satisfies\n\n(a) \( \parallel v\parallel \geq 0 \) for all \( v \) in \( V \), with equality if and only if \( v = 0 \) ,\n\n(b) \( \parallel {cv}\parallel = \left| c\right| \parallel v\parallel \) for all \( v \) in \( V \) and all scalars \( c \) ,\n\n(c) ... | Proof. Conclusion (a) is immediate from properties (iv) and (v) of an inner product, and (b) follows since \( \parallel {cv}{\parallel }^{2} = \left( {{cv},{cv}}\right) = c\bar{c}\left( {v, v}\right) = {\left| c\right| }^{2}\parallel v{\parallel }^{2} \) . Finally we use the law of cosines and the Schwarz inequality (P... | Yes |
Proposition 3.3. Let \( V \) be an inner-product space. If \( \left\{ {{u}_{1},\ldots ,{u}_{k}}\right\} \) is an orthonormal set in \( V \) and if \( v \) is given in \( V \), then there exists a unique decomposition\n\n\[ v = {c}_{1}{u}_{1} + \cdots + {c}_{k}{u}_{k} + {v}^{ \bot } \]\n\nwith \( {v}^{ \bot } \) orthogo... | Proof of UNIQUENESS. Taking the inner product of both sides with \( {u}_{j} \), we obtain \( \left( {v,{u}_{j}}\right) = \left( {{c}_{1}{u}_{1} + \cdots + {c}_{k}{u}_{k} + {v}^{ \bot },{u}_{j}}\right) = {c}_{j} \) for each \( j \) . Then \( {c}_{j} = \left( {v,{u}_{j}}\right) \) is forced, and \( {v}^{ \bot } \) must b... | Yes |
Corollary 3.4 (Bessel’s inequality). Let \( V \) be an inner-product space. If \( \left\{ {{u}_{1},\ldots ,{u}_{k}}\right\} \) is an orthonormal set in \( V \) and if \( v \) is given in \( V \), then \( \mathop{\sum }\limits_{{j = 1}}^{k}{\left| \left( v,{u}_{j}\right) \right| }^{2} \) \( \leq \parallel v{\parallel }^... | Proof. Using Proposition 3.3, write \( v = \mathop{\sum }\limits_{{j = 1}}^{k}\left( {v,{u}_{j}}\right) {u}_{j} + {v}^{ \bot } \) with \( {v}^{ \bot } \) orthogonal to \( {u}_{1},\ldots ,{u}_{k} \) . Then\n\n\[ \parallel v{\parallel }^{2} = \left( {\mathop{\sum }\limits_{{i = 1}}^{k}\left( {v,{u}_{i}}\right) {u}_{i} + ... | Yes |
Proposition 3.5. If \( \left\{ {{v}_{1},\ldots ,{v}_{k}}\right\} \) is a linearly independent set in an inner-product space \( V \), then the Gram-Schmidt orthogonalization process replaces \( \left\{ {{v}_{1},\ldots ,{v}_{k}}\right\} \) by an orthonormal set \( \left\{ {{u}_{1},\ldots ,{u}_{k}}\right\} \) such that \(... | Proof. We argue by induction on \( j \) . The base case is \( j = 1 \), and the result is evident in this case. Assume inductively that \( {u}_{1},\ldots ,{u}_{j - 1} \) are well defined and orthonormal and that \( \operatorname{span}\left\{ {{v}_{1},\ldots ,{v}_{j - 1}}\right\} = \operatorname{span}\left\{ {{u}_{1},\l... | Yes |
Corollary 3.6. If \( V \) is a finite-dimensional inner-product space, then any orthonormal set in a vector subspace \( S \) of \( V \) can be extended to an orthonormal basis of \( S \) . | Proof. Extend the given orthonormal set to a basis of \( S \) by Corollary 2.3b. Then apply the Gram-Schmidt orthogonalization process. The given vectors do not get changed by the process, as we see from the formulas for the vectors \( {u}_{j}^{\prime } \) and \( {u}_{j} \), and hence the result is an extension of the ... | Yes |
Corollary 3.7. If \( S \) is a vector subspace of a finite-dimensional inner-product space \( V \), then \( S \) has an orthonormal basis. | Proof. This is the special case of Corollary 3.6 in which the given orthonormal set is empty. | No |
Theorem 3.8 (Projection Theorem). If \( S \) is a vector subspace of the finite-dimensional inner-product space \( V \), then every \( v \) in \( V \) decomposes uniquely as \( v = {v}_{1} + {v}_{2} \) with \( {v}_{1} \) in \( S \) and \( {v}_{2} \) in \( {S}^{ \bot } \) . In other words, \( V = S \oplus {S}^{ \bot } \... | Proof. Uniqueness follows from the fact that \( S \cap {S}^{ \bot } = 0 \) . For existence, use of Corollaries 3.7 and 3.6 produces an orthonormal basis \( \left\{ {{u}_{1},\ldots ,{u}_{r}}\right\} \) of \( S \) and extends it to an orthonormal basis \( \left\{ {{u}_{1},\ldots ,{u}_{n}}\right\} \) of \( V \) . The vect... | Yes |
Corollary 3.9. If \( S \) is a vector subspace of the finite-dimensional inner-product space \( V \), then\n\n(a) \( \dim V = \dim S + \dim {S}^{ \bot } \) ,\n\n(b) \( {S}^{ \bot \bot } = S \) . | Proof. Conclusion (a) is immediate from the direct-sum decomposition \( V = \) \( S \oplus {S}^{ \bot } \) of Theorem 3.8. For (b), the definition of orthogonal complement gives \( S \subseteq {S}^{ \bot \bot } \) . On the other hand, application of (a) twice shows that \( S \) and \( {S}^{ \bot \bot } \) have the same... | Yes |
Corollary 3.10. Let \( V \) be a finite-dimensional inner-product space, let \( S \) be a vector subspace of \( V \), let \( \left\{ {{u}_{1},\ldots ,{u}_{k}}\right\} \) be an orthonormal basis of \( S \), and let \( E \) be the orthogonal projection of \( V \) on \( S \) . If \( v \) is in \( V \), then\n\n\[ E\left( ... | Proof. Write \( v = \mathop{\sum }\limits_{{j = 1}}^{k}\left( {v,{u}_{j}}\right) {u}_{j} + {v}^{ \bot } \) as in Proposition 3.3. Then each \( {u}_{j} \) is in \( S \), and the vector \( {v}^{ \bot } \), being orthogonal to each member of a basis of \( S \), is in \( {S}^{ \bot } \) . This proves the formula for \( E\l... | Yes |
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