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Theorem 3.11 (Parseval’s equality). If \( V \) is a finite-dimensional inner-product space, then the following conditions on an orthonormal set \( \left\{ {{u}_{1},\ldots ,{u}_{m}}\right\} \) are equivalent:\n\n(a) \( \left\{ {{u}_{1},\ldots ,{u}_{m}}\right\} \) is a vector-space basis of \( V \), hence an orthonormal ... | Proof. Let \( S = \operatorname{span}\left\{ {{u}_{1},\ldots ,{u}_{m}}\right\} \), and let \( E \) be the orthogonal projection of \( V \) on \( S \) . If (a) holds, then \( S = V \) and \( {S}^{ \bot } = 0 \) . Thus (b) holds.\n\nIf (b) holds, then \( {S}^{ \bot } = 0 \) and \( E \) is the identity. Thus (c) holds by ... | Yes |
Theorem 3.12 (Riesz Representation Theorem). If \( \ell \) is a linear functional on the finite-dimensional inner-product space \( V \), then there exists a unique \( v \) in \( V \) with \( \ell \left( u\right) = \left( {u, v}\right) \) for all \( u \) in \( V \) . | Proof. Uniqueness is immediate by subtracting two such expressions, since if \( \left( {u, v}\right) = 0 \) for all \( u \), then the special case \( u = v \) gives \( \left( {v, v}\right) = 0 \) and \( v = 0 \) . Let us prove existence. If \( \ell = 0 \), take \( v = 0 \) . Otherwise let \( S = \ker \ell \) . Corollar... | Yes |
Proposition 3.13. Let \( L : V \rightarrow V \) be a linear map on the finite-dimensional inner-product space \( V \) . For each \( u \) in \( V \), there exists a unique vector \( {L}^{ * }\left( u\right) \) in \( V \) such that\n\n\[ \left( {L\left( v\right), u}\right) = \left( {v,{L}^{ * }\left( u\right) }\right) \;... | Proof. The function \( v \mapsto \left( {L\left( v\right), u}\right) \) is a linear functional on \( V \), and Theorem 3.12 shows that it is given by the inner product with a unique vector of \( V \) . Thus we define \( {L}^{ * }\left( u\right) \) to be the unique vector of \( V \) with \( \left( {L\left( v\right), u}\... | Yes |
Lemma 3.14. If \( L : V \rightarrow V \) is a linear map on the finite-dimensional inner-product space \( V \) and if \( \Gamma = \left( {{u}_{1},\ldots ,{u}_{n}}\right) \) and \( \Delta = \left( {{v}_{1},\ldots ,{v}_{n}}\right) \) are ordered orthonormal bases of \( V \), then the the matrix \( A = \left( \begin{matri... | Proof. Applying Theorem 3.11c, we have\n\n\[ \n{A}_{ij} = {\left( \begin{matrix} L\left( {u}_{j}\right) \\ \Delta \end{matrix}\right) }_{i} = {\left( \begin{matrix} \mathop{\sum }\limits_{{i}^{\prime }}\left( {L\left( {u}_{j}\right) ,{v}_{{i}^{\prime }}}\right) {v}_{{i}^{\prime }} \\ \Delta \end{matrix}\right) }_{i} \n... | Yes |
Proposition 3.15. If \( L : V \rightarrow V \) is a linear map on the finite-dimensional inner-product space \( V \) and if \( \Gamma = \left( {{u}_{1},\ldots ,{u}_{n}}\right) \) and \( \Delta = \left( {{v}_{1},\ldots ,{v}_{n}}\right) \) are ordered orthonormal bases of \( V \), then the matrices \( A = \left( \begin{m... | Proof. Lemma 3.14 and the definition of \( {L}^{ * } \) give \( {A}_{ij}^{ * } = \left( {{L}^{ * }\left( {v}_{j}\right) ,{u}_{i}}\right) = \) \( \left( {{v}_{j}, L\left( {u}_{i}\right) }\right) = \overline{\left( L\left( {u}_{i}\right) ,{v}_{j}\right) } = \overline{{A}_{ji}}. \) | Yes |
Proposition 3.16. If \( V \) is a finite-dimensional inner-product space and \( S \) is a vector subspace of \( V \), then the orthogonal projection \( E : V \rightarrow V \) of \( V \) on \( S \) is self-adjoint. | Proof. Let \( v = {v}_{1} + {v}_{2} \) and \( u = {u}_{1} + {u}_{2} \) be the decompositions of two members of \( V \) according to \( V = S \oplus {S}^{ \bot } \) . Then we have \( \left( {v,{E}^{ * }\left( u\right) }\right) = \left( {E\left( v\right), u}\right) = \) \( \left( {{v}_{1},{u}_{1} + {u}_{2}}\right) = \lef... | Yes |
If \( V \) is a finite-dimensional inner-product space and \( L : V \rightarrow V \) is a self-adjoint linear map, then \( \left( {L\left( v\right), v}\right) \) is in \( \mathbb{R} \) for every \( v \) in \( V \), and consequently every eigenvalue of \( L \) is in \( \mathbb{R} \) . Conversely if \( \mathbb{F} = \math... | Proof. If \( L = {L}^{ * } \), then \( \left( {L\left( v\right), v}\right) = \left( {v,{L}^{ * }\left( v\right) }\right) = \left( {v, L\left( v\right) }\right) = \overline{\left( L\left( v\right), v\right) } \) , and hence \( \left( {L\left( v\right), v}\right) \) is real-valued. If \( v \) is an eigenvector with eigen... | Yes |
Proposition 3.18. If \( V \) is a finite-dimensional inner-product space, then the following conditions on a linear map \( L : V \rightarrow V \) are equivalent:\n\n(a) \( {L}^{ * }L = I \) ,\n\n(b) \( L \) carries some orthonormal basis of \( V \) to an orthonormal basis,\n\n(c) \( L \) carries each orthonormal basis ... | Proof. We prove that (a), (d), and (e) are equivalent and that (b), (c), and (d) are equivalent.\n\nIf (a) holds and \( u \) and \( v \) are given in \( V \), then \( \left( {L\left( u\right), L\left( v\right) }\right) = \left( {{L}^{ * }L\left( u\right), v}\right) = \) \( \left( {I\left( u\right), v}\right) = \left( {... | Yes |
Proposition 3.19. If \( V \) is a finite-dimensional inner-product space, if \( \Gamma = \) \( \left( {{u}_{1},\ldots ,{u}_{n}}\right) \) and \( \Delta = \left( {{v}_{1},\ldots ,{v}_{n}}\right) \) are ordered orthonormal bases of \( V \), and if \( L : V \rightarrow V \) is a linear map that is orthogonal if \( \mathbb... | Proof. Proposition 3.15 and Theorem 2.16 give \( {A}^{ * }A = \left( \begin{matrix} {L}^{ * } \\ {\Gamma \Delta } \end{matrix}\right) \left( \begin{matrix} L \\ {\Delta \Gamma } \end{matrix}\right) = \) \( \left( \begin{matrix} I \\ {\Delta \Delta } \end{matrix}\right) \), and the right side is the identity matrix, as ... | Yes |
Lemma 3.20. If \( L : V \rightarrow V \) is a self-adjoint linear map on an inner-product space \( V \), then \( v \mapsto \left( {L\left( v\right), v}\right) \) is real-valued, every eigenvalue of \( L \) is real, eigenvalues under \( L \) for distinct eigenvalues are orthogonal, and every vector subspace \( S \) of \... | Proof. The first two conclusions are contained in Proposition 3.17. If \( {v}_{1} \) and \( {v}_{2} \) are eigenvectors of \( L \) with distinct real eigenvalues \( {\lambda }_{1} \) and \( {\lambda }_{2} \), then\n\n\[ \left( {{\lambda }_{1} - {\lambda }_{2}}\right) \left( {{v}_{1},{v}_{2}}\right) = \left( {{\lambda }... | Yes |
Corollary 3.22. Let \( L : V \rightarrow V \) be a positive semidefinite linear map on a finite-dimensional inner-product space, and let \( A \) be an \( n \) -by- \( n \) Hermitian matrix. Then\n\n(a) \( L \) or \( A \) is positive semidefinite if and only if all of its eigenvalues are \( \geq 0 \) . | Proof. We apply the Spectral Theorem (Theorem 3.21). For each conclusion the result for a matrix \( A \) is a special case of the result for the linear map \( L \), and it is enough to treat only \( L \) . In (a), let \( \left( {{u}_{1},\ldots ,{u}_{n}}\right) \) be an ordered basis of eigenvectors with respective eige... | Yes |
Let \( V \) be a finite-dimensional inner-product space, and let \( {L}_{1},\ldots ,{L}_{m} \) be self-adjoint linear maps from \( V \) to \( V \) that commute in the sense that \( {L}_{i}{L}_{j} = {L}_{j}{L}_{i} \) for all \( i \) and \( j \) . Then \( V \) has an orthonormal basis of simultaneous eigenvectors of \( {... | This follows by iterating the Spectral Theorem (Theorem 3.21). In fact, let \( \left\{ {V}_{{\lambda }_{1}}\right\} \) be the system of vector subspaces produced by the theorem for \( {L}_{1} \) . For each \( j \), the commutativity of the linear maps \( {L}_{i} \) forces\n\n\[ \n{L}_{1}\left( {{L}_{i}\left( v\right) }... | Yes |
Corollary 3.24. Suppose that \( \mathbb{F} = \mathbb{C} \), and let \( L : V \rightarrow V \) be a normal linear map on the finite-dimensional inner-product space \( V \) . Then \( V \) has an orthonormal basis of eigenvectors of \( L \) . In addition, for each complex scalar \( \lambda \), let \[ {V}_{\lambda } = \{ v... | Proof. The point is that \( L = \left( {\frac{1}{2}\left( {L + {L}^{ * }}\right) }\right) + i\left( {\frac{1}{2i}\left( {L - {L}^{ * }}\right) }\right) \) and that \( \frac{1}{2}\left( {L + {L}^{ * }}\right) \) and \( \frac{1}{2i}\left( {L - {L}^{ * }}\right) \) are self-adjoint. If \( L \) commutes with \( {L}^{ * } \... | Yes |
Corollary 3.25. Suppose that \( \mathbb{F} = \mathbb{C} \), and let \( L : V \rightarrow V \) be a unitary linear map on the finite-dimensional inner-product space \( V \) . Then \( V \) has an orthonormal basis of eigenvectors of \( L \) . In addition, for each complex scalar \( \lambda \), let\n\n\[ \n{V}_{\lambda } ... | Proof. This is a special case of Corollary 3.24 since a unitary linear map \( L \) has \( L{L}^{ * } = I = {L}^{ * }L \) . The eigenvalues all have absolute value 1 as a consequence of Proposition 3.18e. | Yes |
Theorem 3.26 (polar decomposition). If \( L : V \rightarrow V \) is a linear map on a finite-dimensional inner-product space, then \( L \) decomposes as \( L = {UP} \), where \( P \) is positive semidefinite and \( U \) is orthogonal if \( \mathbb{F} = \mathbb{R} \) and unitary if \( \mathbb{F} = \mathbb{C} \). The lin... | Proof of uniqueness. Let \( L = {UP} = {U}^{\prime }{P}^{\prime } \). Then \( {L}^{ * }L = {P}^{2} = {P}^{\prime 2} \). The linear map \( {L}^{ * }L \) is positive semidefinite since its adjoint is \( {\left( {L}^{ * }L\right) }^{ * } = {L}^{ * }{L}^{* * } = {L}^{ * }L \) and since \( \left( {{L}^{ * }L\left( v\right),... | Yes |
Proposition 4.1. Let \( \theta \) be in \( \mathbb{C} \), and suppose for some integer \( n \geq 1 \) that the set \( \left\{ {1,\theta ,{\theta }^{2},\ldots ,{\theta }^{n}}\right\} \) is linearly dependent over \( \mathbb{Q} \). Then the finite-dimensional rational vector space \( \mathbb{Q}\left\lbrack \theta \right\... | Proof. We have seen that \( \mathbb{Q}\left\lbrack \theta \right\rbrack \) is closed under multiplication. If \( x \) is a nonzero member of \( \mathbb{Q}\left\lbrack \theta \right\rbrack \), then all positive powers of \( x \) must be in \( \mathbb{Q}\left\lbrack \theta \right\rbrack \), and the fact that \( \dim \mat... | Yes |
Proposition 4.2 (Cayley’s Theorem). Any group \( G \) is isomorphic to a subgroup of invertible functions on a set \( X \) . The set \( X \) can be taken to be \( G \) itself. In particular any finite group with \( n \) elements is isomorphic to a subgroup of the symmetric group \( {\mathfrak{S}}_{n} \) . | Proof. Define \( X = G \), put \( {f}_{a}\left( x\right) = {ax} \) for \( a \) in \( G \), and let \( {G}^{\prime } = \left\{ {{f}_{a} \mid a \in G}\right\} \) . To see that \( {G}^{\prime } \) is a group, we need \( {G}^{\prime } \) to contain the identity and to be closed under composition and inverses. Since \( {f}_... | Yes |
Proposition 4.3. Each cyclic group \( G \) is isomorphic either to the additive group \( \mathbb{Z} \) of integers or to the additive group \( \mathbb{Z}/m\mathbb{Z} \) of integers modulo \( m \) for some positive integer \( m \) . | Proof. If all \( {a}^{n} \) are distinct, then the rule \( {a}^{m + n} = {a}^{m}{a}^{n} \) implies that the function \( n \mapsto {a}^{n} \) is an isomorphism of \( \mathbb{Z} \) with \( G \) . On the other hand, if \( {a}^{k} = {a}^{l} \) with \( k > l \), then \( {a}^{k - l} = 1 \) and there exists a positive integer... | Yes |
Proposition 4.5. If \( G \) is a group and \( {G}_{1} \) and \( {G}_{2} \) are subgroups, then the following conditions are equivalent:\n\n(a) \( G \) is the internal direct product of \( {G}_{1} \) and \( {G}_{2} \),\n\n(b) every element in \( G \) decomposes uniquely as a product \( {g}_{1}{g}_{2} \) with \( {g}_{1} ... | Proof. We have seen that (a) implies (b). If (b) holds and \( g \) is in \( {G}_{1} \cap {G}_{2} \) , then the formula \( 1 = g{g}^{-1} \) and the uniqueness of the decomposition of 1 as a product together imply that \( g = 1 \) . Hence (c) holds.\n\nIf (c) holds, define \( \varphi : {G}_{1} \times {G}_{2} \rightarrow ... | Yes |
Lemma 4.6. If \( H \) is a subgroup of the group \( G \), then any two left cosets of \( H \) in \( G \) have the same cardinality, namely card \( H \) . | Proof. If \( {g}_{1}H \) and \( {g}_{2}H \) are given, then the map \( g \mapsto {g}_{2}{g}_{1}^{-1}g \) is one-one on \( G \) and carries \( {g}_{1}H \) onto \( {g}_{2}H \) . Hence \( {g}_{1}H \) and \( {g}_{2}H \) have the same cardinality. Taking \( {g}_{1} = 1 \), we see that this common cardinality is card \( H \)... | Yes |
Theorem 4.7 (Lagrange’s Theorem). If \( G \) is a finite group, then \( \left| G\right| = \) \( \left| {G/H}\right| \left| H\right| \) . Consequently the order of any subgroup of \( G \) divides the order of \( G \) . | Proof. Lemma 4.6 shows that each left coset has \( \left| H\right| \) elements. The left cosets are disjoint and exhaust \( G \), and there are \( \left| {G/H}\right| \) left cosets. Thus \( G \) has \( \left| {G/H}\right| \left| H\right| \) elements. | Yes |
Corollary 4.8. If \( G \) is a finite group, then each element \( a \) of \( G \) has finite order, and the order of \( a \) divides the order of \( G \) . | Proof. The order of \( a \) equals \( \left| H\right| \) if \( H = \left\{ {{a}^{n} \mid n \in \mathbb{Z}}\right\} \), and Corollary 4.8 is thus a special case of Theorem 4.7. | No |
Corollary 4.9. If \( p \) is a prime, then the only group of order \( p \), up to isomorphism, is the cyclic group \( {C}_{p} \), and it has no subgroups other than \( \{ 1\} \) and \( {C}_{p} \) itself. | Proof. Suppose that \( G \) is a finite group of order \( p \) and that \( H \neq \{ 1\} \) is a subgroup of \( G \) . Let \( a \neq 1 \) be in \( H \), and let \( P = \left\{ {{a}^{n} \mid n \in \mathbb{Z}}\right\} \) . Since \( a \neq 1 \) , Corollary 4.8 shows that the order of \( a \) is an integer \( > 1 \) that d... | Yes |
Proposition 4.10. If \( H \) is a normal subgroup of a group \( G \), then \( G/H \) becomes a group under the inherited multiplication \( \left( {{g}_{1}H}\right) \left( {{g}_{2}H}\right) = \left( {{g}_{1}{g}_{2}}\right) H \), and the function \( q : G \rightarrow G/H \) given by \( q\left( g\right) = {gH} \) is a hom... | Proof. The coset \( {1H} \) is the identity, and \( {\left( gH\right) }^{-1} = {g}^{-1}H \) . Also, the computation \( \left( {{g}_{1}H{g}_{2}H}\right) {g}_{3}H = {g}_{1}{g}_{2}{g}_{3}H = {g}_{1}H\left( {{g}_{2}H{g}_{3}H}\right) \) proves associativity. Certainly \( q \) is onto \( G/H \) . It is a homomorphism since \... | Yes |
Proposition 4.11. Let \( \varphi : {G}_{1} \rightarrow {G}_{2} \) be a homomorphism between groups, let \( {H}_{0} = \ker \varphi \), let \( H \) be a normal subgroup of \( {G}_{1} \) contained in \( {H}_{0} \), and define \( q : {G}_{1} \rightarrow {G}_{1}/H \) to be the quotient homomorphism. Then there exists a homo... | Proof. We will have \( \bar{\varphi } \circ q = \varphi \) if and only if \( \bar{\varphi } \) satisfies \( \bar{\varphi }\left( {{g}_{1}H}\right) = \varphi \left( {g}_{1}\right) \) . What needs proof is that \( \bar{\varphi } \) is well defined. Thus suppose that \( {g}_{1} \) and \( {g}_{1}^{\prime } \) are in the sa... | Yes |
Corollary 4.12. Let \( \varphi : {G}_{1} \rightarrow {G}_{2} \) be a homomorphism between groups, and suppose that \( \varphi \) is onto \( {G}_{2} \) and has kernel \( H \) . Then \( \varphi \) exhibits the group \( {G}_{1}/H \) as canonically isomorphic to \( {G}_{2} \) . | Proof. Take \( H = {H}_{0} \) in Proposition 4.11, and form \( \bar{\varphi } : {G}_{1}/H \rightarrow {G}_{2} \) with \( \varphi = \bar{\varphi } \circ q \) . The proposition shows that \( \bar{\varphi } \) is onto \( {G}_{2} \) and has trivial kernel, i.e., the identity element of \( {G}_{1}/H \) . Having trivial kern... | Yes |
Theorem 4.13 (First Isomorphism Theorem). Let \( \varphi : {G}_{1} \rightarrow {G}_{2} \) be a homomorphism between groups, and suppose that \( \varphi \) is onto \( {G}_{2} \) and has kernel \( K \) . Then the map \( {H}_{1} \mapsto \varphi \left( {H}_{1}\right) \) gives a one-one correspondence between\n\n(a) the sub... | Proof. The passage from (a) to (b) is by direct image under \( \varphi \), and the passage from (b) to (a) will be by inverse image under \( {\varphi }^{-1} \) . Certainly the direct image of a subgroup as in (a) is a subgroup as in (b). To prove the one-one correspondence, we are to show that the inverse image of a su... | Yes |
Theorem 4.14 (Second Isomorphism Theorem). Let \( {H}_{1} \) and \( {H}_{2} \) be subgroups of a group \( G \) with \( {H}_{2} \) normal in \( G \) . Then \( {H}_{1} \cap {H}_{2} \) is a normal subgroup of \( {H}_{1} \), the set \( {H}_{1}{H}_{2} \) of products is a subgroup of \( G \) with \( {H}_{2} \) as a normal su... | Proof. The set \( {H}_{1} \cap {H}_{2} \) is a subgroup, being the intersection of two subgroups. For \( {h}_{1} \) in \( {H}_{1} \), we have \( {h}_{1}\left( {{H}_{1} \cap {H}_{2}}\right) {h}_{1}^{-1} \subseteq {h}_{1}{H}_{1}{h}_{1}^{-1} \subseteq {H}_{1} \) since \( {H}_{1} \) is a subgroup and \( {h}_{1}\left( {{H}_... | Yes |
Proposition 4.15 (universal mapping property of external direct product). Let \( \\left\\{ {{G}_{s} \\mid s \\in S}\\right\\} \) be a nonempty set of groups, and let \( \\mathop{\\prod }\\limits_{{s \\in S}}{G}_{s} \) be the external direct product, the associated group homomorphisms being the coordinate mappings \( {p... | Proof. Existence of \( \\varphi \) is proved by taking \( \\varphi \\left( h\\right) = {\\left\\{ {\\varphi }_{s}\\left( h\\right) \\right\\} }_{s \\in S} \) . Then \( {p}_{{s}_{0}}\\left( {\\varphi \\left( h\\right) }\\right) \) \( = {p}_{{s}_{0}}\\left( {\\left\\{ {\\varphi }_{s}\\left( h\\right) \\right\\} }_{s \\in... | Yes |
Let \( S \) be a nonempty set of groups, and let \( {G}_{s} \) be the group corresponding to the member \( s \) of \( S \) . If \( \left( {G,\left\{ {p}_{s}\right\} }\right) \) and \( \left( {{G}^{\prime },\left\{ {p}_{s}^{\prime }\right\} }\right) \) are two direct products, then the homomorphisms \( {p}_{s} : G \righ... | Proof. In Figure 4.3 let \( H = {G}^{\prime } \) and \( {\varphi }_{s} = {p}_{s}^{\prime } \) . If \( \Phi : {G}^{\prime } \rightarrow G \) is the homomorphism produced by the fact that \( G \) is a direct product, then we have \( {p}_{s} \circ \Phi = {p}_{s}^{\prime } \) for all \( s \) . Reversing the roles of \( G \... | Yes |
Proposition 4.17 (universal mapping property of external direct sum). Let \( \left\{ {{G}_{s} \mid s \in S}\right\} \) be a nonempty set of abelian groups, and let \( {\bigoplus }_{s \in S}{G}_{s} \) be the external direct sum, the associated group homomorphisms being the embedding mappings \( {i}_{{s}_{0}} : {G}_{{s}_... | Proof. Existence of \( \varphi \) is proved by taking \( \varphi \left( {\left\{ {g}_{s}\right\} }_{s \in S}\right) = \mathop{\sum }\limits_{s}{\varphi }_{s}\left( {g}_{s}\right) \) . The sum on the right side is meaningful since the element \( {\left\{ {g}_{s}\right\} }_{s \in S} \) of the direct sum has only finitely... | Yes |
Proposition 4.18. Let \( S \) be a nonempty set of abelian groups, and let \( {G}_{s} \) be the group corresponding to the member \( s \) of \( S \) . If \( \left( {G,\left\{ {i}_{s}\right\} }\right) \) and \( \left( {{G}^{\prime },\left\{ {i}_{s}^{\prime }\right\} }\right) \) are two direct sums, then the homomorphism... | Proof. In Figure 4.5 let \( H = {G}^{\prime } \) and \( {\varphi }_{s} = {i}_{s}^{\prime } \) . If \( \Phi : G \rightarrow {G}^{\prime } \) is the homomorphism produced by the fact that \( G \) is a direct sum, then we have \( \Phi \circ {i}_{s} \) \( = {i}_{s}^{\prime } \) for all \( s \) . Reversing the roles of \( G... | Yes |
Proposition 4.19. If \( R \) is a ring with identity \( {1}_{R} \), then there exists a unique homomorphism of rings \( {\varphi }_{1} : \mathbb{Z} \rightarrow R \) such that \( \varphi \left( 1\right) = {1}_{R} \) . | Proof. The formulas for manipulating exponents of an element in a group, when translated into the additive notation for addition in \( R \), say that \( n \mapsto {nr} \) satisfies \( \left( {m + n}\right) r = {mr} + {nr} \) and \( \left( {mn}\right) r = m\left( {nr}\right) \) for all \( r \) in \( R \) and all integer... | Yes |
Proposition 4.20. If \( I \) is an ideal in a ring \( R \), then a well-defined operation of multiplication is obtained within the additive group \( R/I \) by the definition \( \left( {{r}_{1} + I}\right) \left( {{r}_{2} + I}\right) = {r}_{1}{r}_{2} + I \), and \( R/I \) becomes a ring. If \( R \) has an identity 1, th... | Proof. If we change the representatives of the cosets from \( {r}_{1} \) and \( {r}_{2} \) to \( {r}_{1} + {i}_{1} \) and \( {r}_{2} + {i}_{2} \) with \( {i}_{1} \) and \( {i}_{2} \) in \( I \), then \( \left( {{r}_{1} + {i}_{1}}\right) \left( {{r}_{2} + {i}_{2}}\right) = {r}_{1}{r}_{2} + \left( {{i}_{1}{r}_{1} + {r}_{... | Yes |
Proposition 4.21. Let \( \varphi : {R}_{1} \rightarrow {R}_{2} \) be a homomorphism of rings, let \( {I}_{0} = \) \( \ker \varphi \), let \( I \) be an ideal of \( {R}_{1} \) contained in \( {I}_{0} \), and let \( q : {R}_{1} \rightarrow {R}_{1}/I \) be the quotient homomorphism. Then there exists a homomorphism of rin... | Proof. Proposition 4.11 shows that \( \varphi \) descends to a homomorphism \( \bar{\varphi } \) of the additive group of \( {R}_{1}/I \) into the additive group of \( {R}_{2} \) and that all the other conclusions hold except possibly for the fact that \( \bar{\varphi } \) respects multiplication. To see that \( \bar{\... | Yes |
Corollary 4.22. Let \( \varphi : {R}_{1} \rightarrow {R}_{2} \) be a homomorphism of rings, and suppose that \( \varphi \) is onto \( {R}_{2} \) and has kernel \( I \) . Then \( \varphi \) exhibits the ring \( {R}_{1}/I \) as canonically isomorphic to \( {R}_{2} \) . | Proof. Take \( I = {I}_{0} \) in Proposition 4.21, and form \( \bar{\varphi } : {R}_{1}/I \rightarrow {R}_{2} \) with \( \varphi = \bar{\varphi } \circ q \) . The proposition shows that \( \bar{\varphi } \) is onto \( {R}_{2} \) and has trivial kernel, i.e., the identity element of \( {R}_{1}/I \) . Having trivial kern... | Yes |
Proposition 4.23. Any field \( \mathbb{F} \) contains a subfield isomorphic to the rationals \( \mathbb{Q} \) or to some field \( {\mathbb{F}}_{p} \) with \( p \) prime. | Proof. Proposition 4.19 produces a homomorphism of rings \( {\varphi }_{1} : \mathbb{Z} \rightarrow \mathbb{F} \) with \( {\varphi }_{1}\left( 1\right) = 1 \) . The kernel of \( {\varphi }_{1} \) is an ideal, necessarily of the form \( m\mathbb{Z} \) with \( m \) an integer \( \geq 0 \), and the image of \( {\varphi }_... | Yes |
Proposition 4.24. Let \( R \) be a nonzero commutative ring with identity, and let \( \iota : R \rightarrow R\left\lbrack X\right\rbrack \) be the identification of \( R \) with constant polynomials. If \( T \) is any commutative ring with identity, if \( \varphi : R \rightarrow T \) is a homomorphism of rings sending ... | Proof. Define \( \Phi \left( {{a}_{0},{a}_{1},\ldots ,{a}_{n},0,\ldots }\right) = \varphi \left( {a}_{0}\right) + \varphi \left( {a}_{1}\right) t + \cdots + \varphi \left( {a}_{n}\right) {t}^{n} \) . It is immediate that \( \Phi \) is a homomorphism of rings sending the identity \( \iota \left( 1\right) = \) \( \left( ... | Yes |
Proposition 4.25. Let \( R \) and \( S \) be nonzero commutative rings with identity, let \( {X}^{\prime } \) be an element of \( S \), and suppose that \( {\iota }^{\prime } : R \rightarrow S \) is a one-one ring homomorphism of \( R \) into \( S \) carrying 1 to 1 . Suppose further that \( \left( {S,{\iota }^{\prime ... | Proof. In the universal mapping property for \( S \), take \( T = R\left\lbrack X\right\rbrack ,\varphi = \iota \), and \( t = X \) . The hypothesis gives us a ring homomorphism \( {\Phi }^{\prime } : S \rightarrow R\left\lbrack X\right\rbrack \) with \( {\Phi }^{\prime }\left( 1\right) = 1,{\Phi }^{\prime } \circ {\io... | Yes |
Proposition 4.26. If \( \mathbb{F} \) is a field, then any finite subgroup of the multiplicative group \( {\mathbb{F}}^{ \times } \) is cyclic. | Proof. Let \( C \) be a subgroup of \( {\mathbb{F}}^{ \times } \) of finite order \( n \) . Lagrange’s Theorem (Corollary 4.8) shows that the order of each element of \( C \) divides \( n \) . With \( h \) defined as the maximum order of an element of \( C \), it is enough to show that \( h = n \) . Let \( a \) be an e... | Yes |
Corollary 4.27. The multiplicative group of a finite field is cyclic. | Proof. This is a special case of Proposition 4.26. | Yes |
Proposition 4.29. If \( R \) is an integral domain, then \( R\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \) is an integral domain. | Proof. Let \( P \) and \( Q \) be nonzero homogeneous polynomials with \( \deg P = d \) and \( \deg Q = {d}^{\prime } \) . We are to prove that \( {PQ} \neq 0 \) . We introduce an ordering on the set of all members \( j \) of \( {J}^{n} \), saying \( j = \left( {{j}_{1},\ldots ,{j}_{n}}\right) > {j}^{\prime } = \left( ... | Yes |
Proposition 4.30. Let \( R \) be a nonzero commutative ring with identity, let \( R\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \) be the ring of polynomials in \( n \) indeterminates, and define \( \iota : R \rightarrow R\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \) to be the identification of \( R \) ... | Proof. If \( P\left( {{X}_{1},\ldots ,{X}_{n}}\right) = \mathop{\sum }\limits_{{{j}_{1} \geq 0,\ldots ,{j}_{n} \geq 0}}{a}_{{j}_{1},\ldots ,{j}_{n}}{X}_{1}^{{j}_{1}}\cdots {X}_{n}^{{j}_{n}} \) is the monomial expansion of a member \( P \) of \( R\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \), then \( \Phi \left... | Yes |
If \( R \) is an infinite integral domain, then the ring homomorphism of \( R\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \) to polynomial functions from \( R \times \cdots \times R \) to \( R \), given by evaluation, is one-one. | Proof. We proceed by induction on \( n \), the case \( n = 1 \) being handled by Proposition 4.28. Assume the result for \( n - 1 \) indeterminates. If \( P \neq 0 \) is in \( R\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \), Corollary 4.31 allows us to write\n\n\[ P\left( {{X}_{1},\ldots ,{X}_{n}}\right) = \mat... | Yes |
Proposition 4.33. If \( \mathbb{F} \) is a finite field, then the number of elements in \( \mathbb{F} \) is a power of a prime. | Proof. The characteristic of \( \mathbb{F} \) cannot be 0 since \( \mathbb{F} \) is finite, and hence it is some prime \( p \) . Denote the prime field of \( \mathbb{F} \) by \( {\mathbb{F}}_{p} \) . By restricting the multiplication so that it is defined only on \( {\mathbb{F}}_{p} \times \mathbb{F} \), we make \( \ma... | Yes |
Proposition 4.34. Let \( G \times X \rightarrow X \) be a group action, let \( p \) be in \( X \), and let \( H \) be the isotropy subgroup at \( p \) . Then the map \( G \rightarrow {Gp} \) given by \( g \mapsto {gp} \) descends to a well-defined map \( G/H \rightarrow {Gp} \) that is one-one from \( G/H \) onto the o... | Proof. Let \( \varphi : G \rightarrow {Gp} \) be defined by \( \varphi \left( g\right) = {gp} \) . For \( h \) in \( H = {G}_{p} \) , \( \varphi \left( {gh}\right) = \left( {gh}\right) p = g\left( {hp}\right) = {gp} = \varphi \left( g\right) \) shows that \( \varphi \) descends to a well-defined function \( \bar{\varph... | Yes |
Corollary 4.35. Let \( G \) be a finite group, let \( G \times X \rightarrow X \) be a group action, let \( p \) be in \( X \), and \( {G}_{p} \) be the isotropy group at \( p \), and let \( {Gp} \) be the orbit of \( p \) . Then \( \left| G\right| = \left| {Gp}\right| \left| {G}_{p}\right| \) . | Proof. Proposition 4.34 shows that the action of \( G \) on some \( G/{G}_{p} \) is the most general group action on a single orbit, \( {G}_{p} \) being the isotropy subgroup. Thus the corollary follows from Lagrange’s Theorem (Theorem 4.7) with \( H = {G}_{p} \) and \( G/H = {Gp} \) . | Yes |
Proposition 4.36. Let \( G \) be a finite group, and let \( p \) be the smallest prime dividing the order of \( G \) . If \( H \) is a subgroup of \( G \) of index \( p \), then \( H \) is normal. | Proof. Let \( X = G/H \), and restrict the group action \( G \times X \rightarrow X \) to an action \( H \times X \rightarrow X \) . The subset \( \{ {1H}\} \) is a single orbit under \( H \), and the remaining \( p - 1 \) members of \( G/H \) form a union of orbits. Corollary 4.35 shows that the number of elements in ... | Yes |
Proposition 4.37. If \( G \) is a finite group, then \( \left| G\right| = \left| {C\ell \left( x\right) }\right| \left| {{Z}_{G}\left( x\right) }\right| \) for all \( x \) in \( G \). | Thus \( \left| {C\ell \left( x\right) }\right| \) is always a divisor of \( \left| G\right| \), and it equals 1 if and only if \( x \) is in the center \( {Z}_{G} \) . Let us apply these considerations to groups whose order is a power of a prime. | No |
Corollary 4.38. If \( G \) is a finite group whose order is a positive power of a prime, then the center \( {Z}_{G} \) is not \( \{ 1\} \) . | Proof. Let \( \left| G\right| = {p}^{n} \) with \( p \) prime and with \( n > 0 \) . The conjugacy classes of \( G \) exhaust \( G \), and thus the sum of all \( \left| {C\ell \left( x\right) }\right| \) ’s equals \( \left| G\right| \) . Since \( \left| {C\ell \left( x\right) }\right| = 1 \) if and only if \( x \) is i... | Yes |
Corollary 4.39. If \( G \) is a finite group of order \( {p}^{2} \) with \( p \) prime, then \( G \) is abelian. | Proof. From Corollary 4.38 we see that either \( \left| {Z}_{G}\right| = {p}^{2} \), in which case \( G \) is abelian, or \( \left| {Z}_{G}\right| = p \) . We show that the latter is impossible. If fact, if \( x \) is not in \( {Z}_{G} \), then \( {Z}_{G}\left( x\right) \) is a subgroup of \( G \) that contains \( {Z}_... | Yes |
Corollary 4.40. If \( G \) is a finite group whose order is a positive power \( {p}^{n} \) of a prime \( p \), then there exist normal subgroups \( {G}_{k} \) of \( G \) for \( 0 \leq k \leq n \) such that \( \left| G\right| = {p}^{k} \) for all \( k \leq n \) and such that \( {G}_{k} \subseteq {G}_{k + 1} \) for all \... | Proof. We proceed by induction on \( n \) . The base case of the induction is \( n = 1 \) and is handled by Corollary 4.9. Assume inductively that the result holds for \( n \), and let \( G \) have order \( {p}^{n + 1} \) . Corollary 4.39 shows that \( {Z}_{G} \neq \{ 1\} \) . Any element \( \neq 1 \) in \( {Z}_{G} \) ... | Yes |
Lemma 4.41. Let \( \sigma \) and \( \tau \) be members of the symmetric group \( {\mathfrak{S}}_{n} \). If \( \sigma \) is expressed as the product of disjoint cycles, then \( {\tau \sigma }{\tau }^{-1} \) has the same cycle structure as \( \sigma \), and the expression for \( {\tau \sigma }{\tau }^{-1} \) as the produ... | Proof. Because the conjugate of a product equals the product of the conjugates, it is enough to handle a cycle \( \gamma = \left( \begin{array}{llll} {a}_{1} & {a}_{2} & \cdots & {a}_{n} \end{array}\right) \) appearing in \( \sigma \). The corresponding cycle \( {\gamma }^{\prime } = {\tau \gamma }{\tau }^{-1} \) is as... | Yes |
Proposition 4.42. Let \( H \) be a subgroup of a symmetric group \( {\mathfrak{S}}_{n} \). If \( C\ell \left( x\right) \) denotes a conjugacy class in \( H \), then all members of \( C\ell \left( x\right) \) have the same cycle structure. Conversely if \( H = {\mathfrak{S}}_{n} \), then the conjugacy class of a permuta... | Proof. The first conclusion is immediate from Lemma 4.41. For the second conclusion, let \( \sigma \) and \( {\sigma }^{\prime } \) have the same cycle structure, and let \( \tau \) be the permutation that moves, for each \( k \), the \( {k}^{\text{th }} \) symbol appearing in the disjoint-cycle expansion of \( \sigma ... | Yes |
Proposition 4.43. Let \( G \) and \( H \) be groups, and let \( \tau \) be a group action of \( G \) on \( H \) by automorphisms. Then the set-theoretic product \( G \times H \) becomes a group \( G{ \times }_{\tau }H \) under the definitions\n\n\[ \left( {{g}_{1},{h}_{1}}\right) \left( {{g}_{2},{h}_{2}}\right) = \left... | Proof. For associativity we compute directly that\n\n\[ \left( {\left( {{g}_{1},{h}_{1}}\right) \left( {{g}_{2},{h}_{2}}\right) }\right) \left( {{g}_{3},{h}_{3}}\right) = \left( {{g}_{1}{g}_{2}{g}_{3},{\tau }_{{g}_{3}^{-1}}\left( {{\tau }_{{g}_{2}^{-1}}\left( {h}_{1}\right) {h}_{2}}\right) {h}_{3}}\right) \]\n\nand \( ... | Yes |
Proposition 4.44. Let \( S \) be a group, let \( G \) and \( H \) be subgroups with \( H \) normal, and suppose that \( G \cap H = \{ 1\} \) and that every element of \( S \) is the product of an element of \( G \) and an element of \( H \) . For each \( g \in G \), define an automorphism \( {\tau }_{g} \) of \( H \) b... | Proof. Since \( {\tau }_{{g}_{1}{g}_{2}}\left( h\right) = {g}_{1}{g}_{2}h{g}_{2}^{-1}{g}_{1}^{-1} = {g}_{1}{\tau }_{{g}_{2}}\left( h\right) {g}_{1}^{-1} = {\tau }_{{g}_{1}}{\tau }_{{g}_{2}}\left( h\right) \) and since each \( {\tau }_{g} \) is an automorphism of \( H,\tau \) is an action by automorphisms. Proposition 4... | Yes |
Lemma 4.45. If \( p \) is a prime, then the automorphisms of the additive group of the field \( {\mathbb{F}}_{p} \) are the multiplications by the members of the multiplicative group \( {\mathbb{F}}_{p}^{ \times } \), and consequently Aut \( {C}_{p} \) is isomorphic to a cyclic group \( {C}_{p - 1} \) . | Proof. Let us write Aut \( {\mathbb{F}}_{p} \) for the automorphism group of the additive group of \( {\mathbb{F}}_{p} \) . Each function \( {\varphi }_{a} : {\mathbb{F}}_{p} \rightarrow {\mathbb{F}}_{p} \) given by \( {\varphi }_{a}\left( n\right) = {na} \), taken modulo \( p \), is in Aut \( {\mathbb{F}}_{p} \) as a ... | Yes |
Proposition 4.46. If \( p \) and \( q \) are primes with \( p < q \) such that \( p \) divides \( q - 1 \) , then there exists a nonabelian group of order \( {pq} \) . | Proof. Let \( G = {C}_{p} \) with generator \( a \), and let \( H = {C}_{q} \) . Lemma 4.45 shows that Aut \( {C}_{q} \cong {C}_{q - 1} \) . Let \( b \) be a generator of Aut \( {C}_{q} \) . Since \( p \) divides \( q - 1,{b}^{\left( {q - 1}\right) /p} \) has order \( p \) . Then the map \( {a}^{k} \rightarrow {b}^{k\l... | Yes |
Lemma 4.48 (Zassenhaus). Let \( {G}_{1},{G}_{2},{G}_{1}^{\prime } \), and \( {G}_{2}^{\prime } \) be subgroups of a group \( G \) with \( {G}_{1}^{\prime } \subseteq {G}_{1} \) and \( {G}_{2}^{\prime } \subseteq {G}_{2},{G}_{1}^{\prime } \) normal in \( {G}_{1} \), and \( {G}_{2}^{\prime } \) normal in \( {G}_{2} \) . ... | Proof. Let us check that \( \left( {{G}_{1} \cap {G}_{2}^{\prime }}\right) {G}_{1}^{\prime } \) is normal in \( \left( {{G}_{1} \cap {G}_{2}}\right) {G}_{1}^{\prime } \) . Handling conjugation by members of \( {G}_{1} \cap {G}_{2} \) is straightforward: If \( g \) is in \( {G}_{1} \cap {G}_{2} \) , then \( g\left( {{G}... | Yes |
Theorem 4.49 (Schreier). Any two normal series of a group \( G \) have equivalent refinements. | Proof. Let the two normal series be\n\n\[ \n{G}_{m} \supseteq {G}_{m - 1} \supseteq \cdots \supseteq {G}_{1} \supseteq {G}_{0} \]\n\n\[ \n{H}_{n} \supseteq {H}_{n - 1} \supseteq \cdots \supseteq {H}_{1} \supseteq {H}_{0} \]\n\n\( \left( *\right) \)\n\nand define\n\[ \n{G}_{ij} = \left( {{G}_{i} \cap {H}_{j}}\right) {G}... | Yes |
Corollary 4.50 (Jordan-Hölder Theorem). Any two composition series of a group \( G \) are equivalent as normal series. | Proof. Let two composition series be given. Theorem 4.49 says that we can insert terms in each so that the refined series have the same length and are equivalent. Since the given series are composition series, the only way to insert a new term is by repeating some term, and the repetition results in a consecutive quoti... | Yes |
Lemma 4.51. Let \( \varphi : G \rightarrow H \) be a homomorphism of abelian groups. If \( \ker \varphi \) and image \( \varphi \) are finitely generated, then \( G \) is finitely generated. | Proof. Let \( \left\{ {{x}_{1},\ldots ,{x}_{m}}\right\} \) and \( \left\{ {{y}_{1},\ldots ,{y}_{n}}\right\} \) be respective finite sets of generators for \( \ker \varphi \) and image \( \varphi \) . For \( 1 \leq j \leq n \), choose \( {x}_{j}^{\prime } \) in \( G \) with \( \varphi \left( {x}_{j}^{\prime }\right) = {... | Yes |
Proposition 4.52. Any subgroup of a finitely generated abelian group is finitely generated. | Proof. Let \( G \) be finitely generated with a set \( \left\{ {{g}_{1},\ldots ,{g}_{n}}\right\} \) of \( n \) generators, and define \( {G}_{k} = \mathbb{Z}{g}_{1} + \cdots + \mathbb{Z}{g}_{k} \) for \( 1 \leq k \leq n \) . If \( H \) is any subgroup of \( G \), define \( {H}_{k} = H \cap {G}_{k} \) for \( 1 \leq k \l... | Yes |
Theorem 4.53. The number of \( \mathbb{Z} \) summands in a free abelian group of finite rank is independent of the direct-sum decomposition of the group. | We define this number to be the rank of the free abelian group. Actually, \ | No |
Lemma 4.54. If \( G \) is a free abelian group with a finite \( \mathbb{Z} \) basis \( {x}_{1},\ldots ,{x}_{n} \), then any linearly independent subset of \( G \) has \( \leq n \) elements. | Proof. Let \( \left\{ {{y}_{1},\ldots ,{y}_{m}}\right\} \) be a linearly independent set in \( G \) . Since \( \left\{ {{x}_{1},\ldots ,{x}_{n}}\right\} \) is a \( \mathbb{Z} \) basis, we can define an \( m \) -by- \( n \) matrix \( C \) of integers by \( {y}_{i} = \mathop{\sum }\limits_{{j = 1}}^{n}{C}_{ij}{x}_{j} \) ... | Yes |
Theorem 4.56 (Fundamental Theorem of Finitely Generated Abelian Groups). Every finitely generated abelian group is a finite direct sum of cyclic groups. The cyclic groups may be taken to be copies of \( \mathbb{Z} \) and various \( {C}_{{p}^{k}} \) with \( p \) prime, and in this case the cyclic groups are unique up to... | Let us establish the setting for the proof of Theorem 4.56. Let \( G \) be the given group, and say that it has a set of \( n \) generators. Proposition 4.17 produces a homomorphism \( \varphi : {\mathbb{Z}}^{n} \rightarrow G \) that carries the standard generators \( {x}_{1},\ldots ,{x}_{n} \) of \( {\mathbb{Z}}^{n} \... | No |
Lemma 4.57. If \( C \) is an \( m \) -by- \( n \) matrix of integers, then there exist an \( m \) -by- \( m \) matrix \( A \) of integers with determinant \( \pm 1 \) and an \( n \) -by- \( n \) matrix \( B \) of integers with determinant \( \pm 1 \) such that for some \( r \geq 0 \), the nonzero entries of \( D = {ACB... | Proof. Given \( C \), choose \( \left( {i, j}\right) \) with \( \left| {C}_{ij}\right| \neq 0 \) but \( \left| {C}_{ij}\right| \) as small as possible. (If \( C = 0 \), the algorithm terminates.) Possibly by interchanging two rows and/or then two columns (a left multiplication with determinant -1 and then a right multi... | Yes |
Lemma 4.58. Let \( {G}_{1},\ldots ,{G}_{n} \) be abelian groups, and for \( 1 \leq j \leq n \), let \( {H}_{j} \) be a subgroup of \( {G}_{j} \). Then \[ \left( {{G}_{1} \oplus \cdots \oplus {G}_{n}}\right) /\left( {{H}_{1} \oplus \cdots \oplus {H}_{n}}\right) \cong \left( {{G}_{1}/{H}_{1}}\right) \oplus \cdots \oplus ... | Proof. Let \( \varphi : {G}_{1} \oplus \cdots \oplus {G}_{n} \rightarrow \left( {{G}_{1}/{H}_{1}}\right) \oplus \cdots \oplus \left( {{G}_{n}/{H}_{n}}\right) \) be the homomorphism defined by \( \varphi \left( {{g}_{1},\ldots ,{g}_{n}}\right) = \left( {{g}_{1}{H}_{1},\ldots ,{g}_{n}{H}_{n}}\right) \). The mapping \( \v... | Yes |
Proposition 4.60. If \( p \) and \( q \) are primes with \( p < q \), then there exists a nonabelian group of order \( {pq} \) if and only if \( p \) divides \( q - 1 \), and in this case the nonabelian group is unique up to isomorphism. It may be taken to be a semidirect product of the cyclic groups \( {C}_{p} \) and ... | Proof. Existence of a nonabelian group of order \( {pq} \), together with the semidirect-product structure, is established by Proposition 4.46 if \( p \) divides \( q - 1 \) . Let us see uniqueness and the necessity of the condition that \( p \) divide \( q - 1 \) .\n\nIf \( G \) has order \( {pq} \), Theorem 4.59a sho... | Yes |
Proposition 4.61. If \( G \) is a group of order 12, then \( G \) contains a subgroup \( H \) of order 3 and a subgroup \( K \) of order 4, and at least one of them is normal. | Proof. Theorem 4.59a shows that \( H \) may be taken to be a Sylow 3-subgroup and \( K \) may be taken to be a Sylow 2-subgroup. We have to prove that either \( H \) or \( K \) is normal.\n\nSuppose that \( H \) is not normal. Theorem 4.59c shows that the number of Sylow 3-subgroups is of the form \( {3k} + 1 \) and di... | Yes |
Lemma 4.62. If \( P \) is a Sylow \( p \) -subgroup of \( G \) and if \( H \) is a subgroup of the normalizer \( N\left( P\right) \) whose order is a power of \( p \), then \( H \subseteq P \) . | Proof. Since \( H \subseteq N\left( P\right) \) and \( P \) is normal in \( N\left( P\right) \), the set \( {HP} \) of products is a group, by the same argument as used for \( {H}_{p}{H}_{q} \) in the proof of Proposition 4.60. Then \( {HP}/P \cong H/\left( {H \cap P}\right) \) by the Second Isomorphism Theorem (Theore... | Yes |
Proposition 4.63. Let \( \mathcal{C} \) be a category, and let \( S \) be a nonempty set. If \( {\left\{ {X}_{s}\right\} }_{s \in S} \) is an object in \( {\mathcal{C}}^{S} \) and if \( \left( {X,\left\{ {p}_{s}\right\} }\right) \) and \( \left( {{X}^{\prime },\left\{ {p}_{s}^{\prime }\right\} }\right) \) are two produ... | Proof. In Figure 4.11 let \( A = {X}^{\prime } \) and \( {\varphi }_{s} = {p}_{s}^{\prime } \) . If \( \Phi \in \operatorname{Morph}\left( {{X}^{\prime }, X}\right) \) is the morphism produced by the fact that \( X \) is a direct product, then we have \( {p}_{s}\Phi = {p}_{s}^{\prime } \) for all \( s \) . Reversing th... | Yes |
Proposition 4.64. Let \( \mathcal{C} \) be a category, and let \( S \) be a nonempty set. If \( {\left\{ {X}_{s}\right\} }_{s \in S} \) is an object in \( {\mathcal{C}}^{S} \) and if \( \left( {X,\left\{ {i}_{s}\right\} }\right) \) and \( \left( {{X}^{\prime },\left\{ {i}_{s}^{\prime }\right\} }\right) \) are two copro... | Proof. In Figure 4.12 let \( A = {X}^{\prime } \) and \( {\varphi }_{s} = {i}_{s}^{\prime } \) . If \( \Phi \in \operatorname{Morph}\left( {X,{X}^{\prime }}\right) \) is the morphism produced by the fact that \( X \) is a coproduct, then we have \( \Phi {i}_{s} = {i}_{s}^{\prime } \) for all \( s \) . Reversing the rol... | Yes |
Proposition 5.1. If \( R \) is a commutative ring with identity, then the determinant function det : \( {M}_{n}\left( R\right) \rightarrow R \) has the following properties:\n\n(a) \( \det \left( {AB}\right) = \det A\det B \) ,\n\n(b) \( \det I = 1 \) ,\n\n(c) \( \det {A}^{t} = \det A \) ,\n\n(d) \( \det C = \det A + \... | Proof. Part (a) was proved above, and parts (c) through (f) may be proved in the same way from the corresponding facts about integer matrices in Section II.7. Part (b) is immediate from the definition.\n\nFor \( \left( \mathrm{g}\right) \), we first prove the result when the entries are in \( \mathbb{Q} \), and then we... | Yes |
Corollary 5.3 (Vandermonde matrix and determinant). If \( {r}_{1},\ldots ,{r}_{n} \) lie in a commutative ring \( R \) with identity, 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 ... | Proof. The derivation of this from Proposition 5.2 is the same as the derivation of Corollary 2.37 from Proposition 2.35. | No |
Proposition 5.4 (Cramer’s rule). Let \( R \) be a commutative ring with identity, let \( A \) be in \( {M}_{n}\left( R\right) \), and define \( {A}^{\text{adj }} \) in \( {M}_{n}\left( R\right) \) to be the classical adjoint of \( A \) , namely the matrix with entries \( {A}_{ij}^{\mathrm{{adj}}} = {\left( -1\right) }^... | Proof. This may be derived from Proposition 2.38 in the same way as for Propositions 5.1 and 5.2 using the principle of permanence of identities. | No |
Corollary 5.5. Let \( R \) be a commutative ring with identity, and let \( A \) be in \( {M}_{n}\left( R\right) \) . If \( \det A \) is a unit in \( R \), then \( A \) has a two-sided inverse in \( {M}_{n}\left( R\right) \) . Conversely if \( A \) has a one-sided inverse in \( {M}_{n}\left( R\right) \), then \( \det A ... | Proof. If \( \det A \) is a unit in \( R \), let \( r \) be its multiplicative inverse. Then Proposition 5.4 shows that \( {r}^{-1}{A}^{\text{adj }} \) is a two-sided inverse of \( A \) . Conversely if \( A \) has, say, a left inverse \( B \), then \( {BA} = I \) implies \( \left( {\det B}\right) \left( {\det A}\right)... | Yes |
Proposition 5.6. If \( \mathbb{K} \) is an algebraically closed field, if \( V \) is a finite-dimensional vector space over \( \mathbb{K} \), and if \( L : V \rightarrow V \) is linear, then \( V \) has an ordered basis in which the matrix of \( L \) is upper triangular. Consequently any member of \( {M}_{n}\left( \mat... | Proof. We proceed by induction on \( n = \dim V \), with the base case \( n = 1 \) being clear. Suppose that the result holds for all linear maps from spaces of dimension \( < n \) to themselves. Given \( L : V \rightarrow V \) with \( \dim V = n \), let \( {v}_{1} \) be an eigenvector of \( L \) . This exists by the r... | Yes |
Proposition 5.7. If \( A \) is in \( {M}_{n}\left( \mathbb{K}\right) \), then there exists a nonzero polynomial \( P \) in \( \mathbb{K}\left\lbrack \lambda \right\rbrack \) such that \( P\left( A\right) = 0 \) . | Proof. The \( \mathbb{K} \) vector space \( {M}_{n}\left( \mathbb{K}\right) \) has dimension \( {n}^{2} \) . Therefore the \( {n}^{2} + 1 \) matrices \( I, A,{A}^{2},\ldots ,{A}^{{n}^{2}} \) are linearly dependent, and we have\n\n\[ \n{c}_{0} + {c}_{1}A + {c}_{2}{A}^{2} + \cdots + {c}_{{n}^{2}}{A}^{{n}^{2}} = 0 \n\]\n\... | Yes |
Proposition 5.8. If \( I \) is a nonzero ideal in \( \mathbb{K}\left\lbrack \lambda \right\rbrack \), then there exists a unique monic polynomial of lowest degree in \( I \), and every member of \( I \) is the product of this particular polynomial by some other polynomial. | Proof. Let \( B\left( \lambda \right) \) be a nonzero member of \( I \) of lowest possible degree; adjusting \( B \) by a scalar factor, we may assume that \( B \) is monic. If \( A \) is in \( I \), then Proposition 1.12 produces polynomials \( Q \) and \( R \) such that \( A = {BQ} + R \) and either \( R = 0 \) or \(... | Yes |
Theorem 5.9 (Cayley-Hamilton Theorem). If \( A \) is in \( {M}_{n}\left( \mathbb{K}\right) \) and if \( F\left( \lambda \right) = \) \( \det \left( {{\lambda I} - A}\right) \) is its characteristic polynomial, then \( F\left( A\right) = 0 \) . | Proof. Let \( T \) be the commutative subring of \( {M}_{n}\left( \mathbb{K}\right) \) generated by \( \mathbb{K}I \) and \( A \) , and define a member \( B\left( \lambda \right) \) of the ring \( T\left\lbrack \lambda \right\rbrack \) by \( B\left( \lambda \right) = {\lambda I} - A \) . The \( {\left( i, j\right) }^{\... | Yes |
Corollary 5.10. If \( A \) is in \( {M}_{n}\left( \mathbb{K}\right) \), then the minimal polynomial of \( A \) divides the characteristic polynomial of \( A \) . | Proof. Theorem 5.9 shows that the characteristic polynomial of \( A \) lies in the ideal of all polynomials vanishing on \( A \) . Then the corollary follows from Proposition 5.8. | Yes |
Proposition 5.11. Let \( V \) be a finite-dimensional vector space over \( \mathbb{K} \), let \( L : V \rightarrow V \) be linear, let \( U \) be a proper nonzero invariant subspace under \( L \), and let \( \bar{L} : V/U \rightarrow V/U \) be the induced linear map on \( V/U \) . Then the characteristic polynomials of... | Proof. Let \( {\Gamma }_{U} = \left( {{v}_{1},\ldots ,{v}_{k}}\right) \) be an ordered basis of \( U \), and extend \( {\Gamma }_{U} \) to an ordered basis \( \Gamma = \left( {{v}_{1},\ldots ,{v}_{n}}\right) \) of \( V \) . Then \( \bar{\Gamma } = \left( {{v}_{k + 1} + U,\ldots ,{v}_{n} + U}\right) \) is an ordered bas... | Yes |
Proposition 5.12. Let \( V \) be a finite-dimensional vector space over \( \mathbb{K} \), let \( L : V \rightarrow V \) be linear, let \( U \) be a proper nonzero invariant subspace under \( L \), and let \( \bar{L} : V/U \rightarrow V/U \) be the induced linear map on \( V/U \) . Then the minimal polynomials of \( {\l... | Proof. Let \( N\left( \lambda \right) \) be the minimal polynomial of \( {\left. L\right| }_{U} \) . Then \( N\left( \lambda \right) \) is the unique monic polynomial of lowest degree in the ideal of all polynomials \( P\left( \lambda \right) \) such that \( P\left( L\right) u = 0 \) for all \( u \) in \( U \) . The mi... | Yes |
Corollary 5.13. If \( L : V \rightarrow V \) is linear on a finite-dimensional vector space over \( \mathbb{K} \) and if a first-degree polynomial \( \lambda - {\lambda }_{0} \) divides the characteristic polynomial of \( L \), then \( \lambda - {\lambda }_{0} \) divides the minimal polynomial of \( L \) . | Proof. If \( \lambda - {\lambda }_{0} \) divides the characteristic polynomial, then \( {\lambda }_{0} \) is an eigenvalue of \( L \), say with \( v \) as an eigenvector. Then \( U = \mathbb{K}v \) is an invariant subspace under \( L \), and the characteristic and minimal polynomials of \( {\left. L\right| }_{U} \) are... | Yes |
Theorem 5.14. If \( L : V \rightarrow V \) is linear on a finite-dimensional vector space over \( \mathbb{K} \), then \( L \) has a basis of eigenvectors if and only if the minimal polynomial \( M\left( \lambda \right) \) of \( L \) is the product of distinct factors of degree 1 ; in this case, \( M\left( \lambda \righ... | Proof. The easy direction is that \( {v}_{1},\ldots ,{v}_{n} \) are the members of a basis of eigenvectors for \( L \) with respective eigenvalues \( {\mu }_{1},\ldots ,{\mu }_{n} \) . In this case, let \( {\lambda }_{1},\ldots ,{\lambda }_{k} \) be the distinct members of the set of eigenvalues, with \( {\mu }_{i} = {... | Yes |
Proposition 5.15. Two diagonal matrices \( A \) and \( {A}^{\prime } \) in \( {M}_{n}\left( \mathbb{K}\right) \) with respective diagonal entries \( {d}_{1},\ldots ,{d}_{n} \) and \( {d}_{1}^{\prime },\ldots ,{d}_{n}^{\prime } \) are similar if and only if there is a permutation \( \sigma \) in \( {\mathfrak{S}}_{n} \)... | Proof. The respective characteristic polynomials are \( \mathop{\prod }\limits_{{j = 1}}^{n}\left( {\lambda - {d}_{j}}\right) \) and \( \mathop{\prod }\limits_{{j = 1}}^{n}\left( {\lambda - {d}_{j}^{\prime }}\right) \) . If \( A \) and \( {A}^{\prime } \) are similar, then the characteristic polynomials are equal, and ... | Yes |
Proposition 5.16. If \( V \) is a vector space and \( {E}_{1} : V \rightarrow V \) is a linear map such that \( {E}_{1}^{2} = {E}_{1} \), then there exists a direct-sum decomposition \( V = {U}_{1} \oplus {U}_{2} \) such that \( {E}_{1} \) is the projection of \( V \) on \( {U}_{1} \) along \( {U}_{2} \) . In this case... | Proof. Define \( {U}_{1} = \) image \( {E}_{1} \) and \( {U}_{2} = \ker {E}_{1} \) . If \( v \) is in image \( {E}_{1} \cap \ker {E}_{1} \) , then \( {E}_{1}\left( v\right) = 0 \) since \( v \) is in \( \ker {E}_{1} \) and \( v = {E}_{1}\left( w\right) \) for some \( w \) in \( V \) since \( v \) is in image \( {E}_{1}... | Yes |
Proposition 5.17. If \( V \) is a vector space and \( {E}_{j} : V \rightarrow V \) for \( 1 \leq j \leq r \) are linear maps such that\n\n(a) \( {E}_{j}{E}_{i} = 0 \) if \( i \neq j \), and\n\n(b) \( {E}_{1} + \cdots + {E}_{r} = I \) ,\nthen \( {E}_{j}^{2} = {E}_{j} \) for \( 1 \leq j \leq r \) and the vector subspaces... | Proof. Multiplying (b) through by \( {E}_{j} \) on the left and applying (a) to each term on the left side except the \( {j}^{\text{th }} \), we obtain \( {E}_{j}^{2} = {E}_{j} \) . Therefore, for each \( j \) , \( {E}_{j} \) is a projection on \( {U}_{j} \) along some vector subspace depending on \( j \) .\n\nIf \( v ... | Yes |
Proposition 5.18. Suppose that a vector space \( V \) is a direct sum \( V = \) \( {U}_{1} \oplus \cdots \oplus {U}_{r} \) of vector subspaces, that \( {E}_{1},\ldots ,{E}_{r} \) are the corresponding projections, and that \( L : V \rightarrow V \) is linear. Then all the subspaces \( {U}_{j} \) are invariant under \( ... | Proof. If \( L\left( {U}_{j}\right) \subseteq {U}_{j} \) for all \( j \), then \( i \neq j \) implies \( {E}_{i}L\left( {U}_{j}\right) \subseteq {E}_{i}\left( {U}_{j}\right) = 0 \) and \( L{E}_{i}\left( {U}_{j}\right) = L\left( 0\right) = 0 \) . Also, \( v \in {U}_{j} \) implies \( {E}_{j}L\left( v\right) = L\left( v\r... | Yes |
Theorem 5.19 (Primary Decomposition Theorem). Let \( L : V \rightarrow V \) be linear on a finite-dimensional vector space over \( \mathbb{K} \), and let \( M\left( \lambda \right) = {P}_{1}{\left( \lambda \right) }^{{l}_{1}}\cdots {P}_{k}{\left( \lambda \right) }^{{l}_{k}} \) be the unique factorization of the minimal... | Proof. For \( 1 \leq j \leq k \), define \( {Q}_{j}\left( \lambda \right) = M\left( \lambda \right) /{P}_{j}{\left( \lambda \right) }^{{l}_{j}} \) . The ideal in \( \mathbb{K}\left\lbrack \lambda \right\rbrack \) generated by \( {Q}_{1}\left( \lambda \right) ,\ldots ,{Q}_{k}\left( \lambda \right) \) consists of all pro... | Yes |
Theorem 5.21. For any field \( \mathbb{K} \), each nilpotent matrix \( N \) in \( {M}_{n}\left( \mathbb{K}\right) \) is similar to a matrix in Jordan form. | The proof of Theorem 5.21 and of the uniqueness statements in Theorem 5.20 will occupy the remainder of this section. It is implicit in Theorem 5.21 that a nilpotent matrix in \( {M}_{n}\left( \mathbb{K}\right) \) has 0 as a root of its characteristic polynomial with multiplicity \( n \), in particular that the only pr... | No |
Lemma 5.22. If \( N \) is a nilpotent matrix in \( {M}_{n}\left( K\right) \), then \( N \) has characteristic polynomial \( {\lambda }^{n} \) and satisfies \( {N}^{n} = 0 \) . | Proof. If \( {N}^{l} = 0 \), then\n\n\[ \left( {{\lambda I} - N}\right) \left( {{\lambda }^{l - 1}I + {\lambda }^{l - 2}N + \cdots + {\lambda }^{2}{N}^{l - 3} + \lambda {N}^{l - 2} + {N}^{l - 1}}\right) = {\lambda }^{l}I - {N}^{l} = {\lambda }^{l}I.\]\n\nTaking determinants and using Proposition 5.1 in the ring \( R = ... | Yes |
Lemma 5.23. Suppose \( j \geq 1 \) and suppose \( {S}_{j} \) is any vector subspace of \( V \) such that \( {K}_{j + 1} = {K}_{j} \oplus {S}_{j} \) . Then \( N \) is one-one from \( {S}_{j} \) into \( {K}_{j} \) and \( N\left( {S}_{j}\right) \cap {K}_{j - 1} = 0 \) . | Proof. Since \( N\left( {\ker {N}^{j + 1}}\right) \subseteq \ker {N}^{j} \), we obtain \( N\left( {S}_{j}\right) \subseteq {K}_{j} \) ; thus \( N \) indeed sends \( {S}_{j} \) into \( {K}_{j} \) . To see that \( N \) is one-one from \( {S}_{j} \) into \( {K}_{j} \), suppose that \( s \) is a member of \( {S}_{j} \) wit... | Yes |
Lemma 5.24. Define \( {U}_{n} = {W}_{n} = 0 \) . For \( 0 \leq j \leq n - 1 \), there exist vector subspaces \( {U}_{j} \) and \( {W}_{j} \) of \( {K}_{j + 1} \) such that\n\n\[ \n{K}_{j + 1} = {K}_{j} \oplus {U}_{j} \oplus {W}_{j} \n\]\n\n\[ \n{U}_{j} = N\left( {{U}_{j + 1} \oplus {W}_{j + 1}}\right) \n\]\n\nand\n\[ \... | Proof. Define \( {U}_{n - 1} = N\left( {{U}_{n} \oplus {W}_{n}}\right) = 0 \), and let \( {W}_{n - 1} \) be a vector subspace such that \( V = {K}_{n} = {K}_{n - 1} \oplus {W}_{n - 1} \) . Put \( {S}_{n - 1} = {U}_{n - 1} \oplus {W}_{n - 1} \) . Proceeding inductively downward, suppose that \( {U}_{n},{U}_{n - 1},\ldot... | Yes |
Lemma 5.25. The vector subspaces of Lemma 5.24 satisfy\n\n\[ V = {U}_{0} \oplus {W}_{0} \oplus {U}_{1} \oplus {W}_{1} \oplus \cdots \oplus {U}_{n - 1} \oplus {W}_{n - 1}. \]\n | PROOF. Iterated use of Lemma 5.24 gives\n\n\[ V = {K}_{n} = {K}_{n - 1} \oplus \left( {{U}_{n - 1} \oplus {W}_{n - 1}}\right) \]\n\n\[ = {K}_{n - 2} \oplus \left( {{U}_{n - 2} \oplus {W}_{n - 2}}\right) \oplus \left( {{U}_{n - 1} \oplus {W}_{n - 1}}\right) \]\n\n\[ = \cdots = {K}_{0} \oplus \left( {{U}_{0} \oplus {W}_{... | Yes |
Proposition 6.1. Let \( \\langle \\cdot , \\cdot \\rangle \) be a bilinear form on a finite-dimensional vector space \( V \), let \( \\Gamma \) and \( \\Delta \) be ordered bases of \( V \), and let \( B \) and \( C \) be the respective matrices of \( \\langle \\cdot , \\cdot \\rangle \) relative to \( \\Gamma \) and \... | The qualitative conclusion about the matrices may be a little unexpected. It is not that they are similar but that they are related by \( C = {S}^{t}{BS} \) for some nonsingular square matrix \( S \). In particular, \( B \) and \( C \) need not have the same determinant. | Yes |
Proposition 6.2. If \( \\langle \\cdot , \\cdot \\rangle \) is any bilinear form on a finite-dimensional vector space \( V \), then\n\n\[ \n\\dim \\operatorname{lrad}\\left( {\\langle \\;\\cdot \\;,\\; \\cdot \\;\\rangle }\\right) = \\dim \\operatorname{rrad}\\left( {\\langle \\;\\cdot \\;,\\; \\cdot \\;\\rangle }\\rig... | Proof. We saw above that computations with bilinear forms of \( V \) reduce, once we choose an ordered basis for \( V \), to computations with matrices, row vectors, and column vectors. Thus the argument just given in the continuation of Example 1 is completely general, and the proposition is proved. | No |
Proposition 6.3. If \( \\langle \\cdot , \\cdot \\rangle \) is a nondegenerate bilinear form on the finite-dimensional vector space \( V \) and if \( U \) is a vector subspace of \( V \), then\n\n\[ \n\\dim V = \\dim U + \\dim {U}^{ \\bot }.\n\] | Proof. Define \( \\ell : V \\rightarrow {U}^{\\prime } \) by \( \\ell \\left( v\\right) \\left( u\\right) = \\langle v, u\\rangle \) for \( v \\in V \) and \( u \\in U \) . The definition of \( {U}^{ \\bot } \) shows that \( \\ker \\ell = {U}^{ \\bot } \) . To see that image \( \\ell = {U}^{\\prime } \), choose a vecto... | Yes |
Corollary 6.4. If \( \langle \cdot , \cdot \rangle \) is a nondegenerate bilinear form on the finite-dimensional vector space \( V \) and if \( U \) is a vector subspace of \( V \), then \( V = U \oplus {U}^{ \bot } \) if and only if \( \langle \cdot , \cdot {\left. \right| }_{U \times U} \) is nondegenerate. | Proof. Corollary 2.29 and Proposition 6.3 together give\n\n\[ \dim \left( {U + {U}^{ \bot }}\right) + \dim \left( {U \cap {U}^{ \bot }}\right) = \dim U + \dim {U}^{ \bot } = \dim V.\]\n\nThus \( U + {U}^{ \bot } = V \) if and only if \( U \cap {U}^{ \bot } = 0 \), if and only if \( \langle \cdot , \cdot \rangle {\left.... | Yes |
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