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Proposition 26.15. If both \( \left( {{p}_{1},{p}_{2},{p}_{3},{p}_{4}}\right) \) and \( \left( {{q}_{1},{q}_{2},{q}_{3},{q}_{4}}\right) \) satisfy the conditions of Proposition 26.14, then the image of the convex hull of \( \left( {{p}_{1},{p}_{2},{p}_{3},{p}_{4}}\right) \) under the unique projective map mapping \( \l... | Proof. It suffices to show that the restriction of our projective transformation maps a line segment to the convex hull of the images of the endpoints of this segment. Thus, the problem reduces to proving that if a projective transformation given by an invertible matrix \[ \left( \begin{array}{ll} a & b \\ c & d \end{a... | Yes |
Proposition 26.16. Given a vector space \( E \) and a hyperplane \( H \) in \( E \), the complement \( {E}_{H} = \mathbf{P}\left( E\right) - \mathbf{P}\left( H\right) \) of the projective hyperplane \( \mathbf{P}\left( H\right) \) in the projective space \( \mathbf{P}\left( E\right) \) can be given an affine structure ... | Proof. Since \( H \) is a hyperplane in \( E \), there is some \( w \in E - H \) such that \( E = {Kw} \oplus H \) . Thus, every vector \( u \) in \( E - H \) can be written in a unique way as \( {\lambda w} + h \), where \( \lambda \neq 0 \) and \( h \in H \) . As a consequence, for every point \( \left\lbrack u\right... | Yes |
Proposition 26.17. Given any affine space \( \left( {E,\overrightarrow{E}}\right) \), for every projective space \( \mathbf{P}\left( F\right) \) (where \( F \) is some vector space), every hyperplane \( H \) in \( F \), and every map \( f : E \rightarrow \mathbf{P}\left( F\right) \) such that \( f\left( E\right) \subse... | Proof. The existence of \( \widetilde{f} \) is a consequence of Proposition 25.6, where we observe that \( \widehat{{F}_{H}} \) is isomorphic to \( F \) . Just take the projective map \( \mathbf{P}\left( \widehat{f}\right) : \widetilde{E} \rightarrow \mathbf{P}\left( F\right) \), where \( \widehat{f} : \widehat{E} \rig... | Yes |
Proposition 26.18. (Pappus) Given any projective plane \( \mathbf{P}\left( E\right) \) and any two distinct lines \( D \) and \( {D}^{\prime } \), for any distinct points \( a, b, c,{a}^{\prime },{b}^{\prime },{c}^{\prime } \), with \( a, b, c \) on \( D \) and \( {a}^{\prime },{b}^{\prime },{c}^{\prime } \) on \( {D}^... | Proof. First, since any two lines in a projective plane intersect in a single point, the points \( p, q, r \) are well defined. Choose \( \Delta = \langle p, r\rangle \) as the line at infinity, and consider the affine plane \( X = \mathbf{P}\left( E\right) - \Delta \) . Since \( \left\langle {a,{b}^{\prime }}\right\ra... | Yes |
Proposition 26.19. (Desargues) Let \( \mathbf{P}\left( E\right) \) be a projective space. Given two triangles \( \left( {a, b, c}\right) \) and \( \left( {{a}^{\prime },{b}^{\prime },{c}^{\prime }}\right) \), where the points \( a, b, c,{a}^{\prime },{b}^{\prime },{c}^{\prime } \) are pairwise distinct and the lines \(... | Proof. First, it is immediately shown that the line \( \langle p, q\rangle \) is distinct from the lines \( A, B, C \) , \( {A}^{\prime },{B}^{\prime },{C}^{\prime } \) . Let us assume that \( \mathbf{P}\left( E\right) \) has dimension \( n \geq 3 \) . If the seven points \( d, a, b, c,{a}^{\prime },{b}^{\prime },{c}^{... | Yes |
Given any two projective lines \( \Delta \) and \( {\Delta }^{\prime } \), for any sequence \( \left( {a, b, c, d}\right) \) of points in \( \Delta \) and any sequence \( \left( {{a}^{\prime },{b}^{\prime },{c}^{\prime },{d}^{\prime }}\right) \) of points in \( {\Delta }^{\prime } \), if \( a, b, c \) are distinct and ... | First, assume that \( f : \Delta \rightarrow {\Delta }^{\prime } \) is a projectivity such that \( f\left( a\right) = {a}^{\prime }, f\left( b\right) = {b}^{\prime } \) , \( f\left( c\right) = {c}^{\prime } \), and \( f\left( d\right) = {d}^{\prime } \) . Let \( h : \Delta \rightarrow {\mathbb{P}}_{K}^{1} \) be the uni... | Yes |
Given any projective line \( \Delta = \mathbf{P}\left( D\right) \), for any three distinct points \( a, b, c \) in \( \Delta \), if \( a = p\left( u\right), b = p\left( v\right) \), and \( c = p\left( {u + v}\right) \), where \( \left( {u, v}\right) \) is a basis of \( D \), and for any \( {\left\lbrack \lambda ,\mu \r... | Proof. If \( \left( {{e}_{1},{e}_{2}}\right) \) is the basis of \( {K}^{2} \) such that \( {e}_{1} = \left( {1,0}\right) \) and \( {e}_{2} = \left( {0,1}\right) \), it is obvious that \( p\left( {e}_{1}\right) = \infty, p\left( {e}_{2}\right) = 0 \), and \( p\left( {{e}_{1} + {e}_{2}}\right) = 1 \) . Let \( f : D \righ... | No |
Proposition 26.23. Let \( f : E \rightarrow E \) be a bijective linear map of a finite-dimensional vector space \( E \) and assume that \( f \neq \mathrm{{id}} \) and that \( f\left( x\right) = x \) for all \( x \in H \), where \( H \) is some hyperplane in \( E \) . If \( \det \left( f\right) = 1 \), then \( f \) is a... | Proof. Only the last part was not proved in Proposition 8.23, Since \( f \) is bijective and the identity on \( H \), the linear map \( f \) -id has kernel exactly \( H \) . Since \( H \) is a hyperplane in \( E \) , the image of \( f \) - id has dimension 1, and since \( u \) belong to this image, it is uniquely defin... | Yes |
Proposition 26.24. If \( h : \mathbb{P}\left( E\right) \rightarrow \mathbb{P}\left( E\right) \) is a homology of axis \( \mathbb{P}\left( H\right) \) and if \( h \neq \mathrm{{id}} \), then for any linear isomorphism \( f : E \rightarrow E \) such that \( h = \mathbb{P}\left( f\right) \), the following properties hold:... | Proof. Since the restriction of \( h = \mathbb{P}\left( f\right) \) to \( \mathbb{P}\left( H\right) \) is the identity, and since \( \mathbb{P}\left( f\right) = \mathbb{P}\left( {\mathrm{{id}}}_{H}\right) \) , by Proposition 26.4 we have \( f = \lambda {\operatorname{id}}_{H} \) on \( H \) for some nonzero scalar \( \l... | Yes |
Given any two lines \( {D}_{1},{D}_{2} \) in a real Euclidean plane \( \left( {E,\overrightarrow{E}}\right) \), letting \( {D}_{I} \) and \( {D}_{J} \) be the isotropic lines in \( {\widetilde{E}}_{\mathbb{C}} \) joining the intersection point \( {D}_{1} \cap {D}_{2} \) of \( {D}_{1} \) and \( {D}_{2} \) to the circula... | In particular, note that \( \theta = \pi /2 \) iff \( \left\lbrack {{D}_{1},{D}_{2},{D}_{I},{D}_{J}}\right\rbrack = - 1 \), that is, if \( \left( {{D}_{1},{D}_{2},{D}_{I},{D}_{J}}\right) \) forms a harmonic division. Thus, two lines \( {D}_{1} \) and \( {D}_{2} \) are orthogonal iff they form a harmonic division with \... | Yes |
Proposition 27.2. Let \( E \) be a Euclidean space of finite dimension \( n \), and let \( f : E \rightarrow E \) be an isometry. For any subspace \( F \) of \( E \), if \( f\left( F\right) = F \), then \( f\left( {F}^{ \bot }\right) \subseteq {F}^{ \bot } \) and \( E = F \oplus {F}^{ \bot } \) . | Proof. We just have to prove that if \( w \in E \) is orthogonal to every \( u \in F \), then \( f\left( w\right) \) is also orthogonal to every \( u \in F \) . However, since \( f\left( F\right) = F \), for every \( v \in F \), there is some \( u \in F \) such that \( f\left( u\right) = v \), and we have\n\n\[ f\left(... | Yes |
Proposition 27.3. Let \( E \) be a Euclidean space.\n\n(1) If \( E \) has odd dimension \( n = {2m} + 1 \), then every rotation \( f \) admits 1 as an eigenvalue and the eigenspace \( F \) of all eigenvectors left invariant under \( f \) has an odd dimension \( {2p} + 1 \) . Furthermore, there is an orthonormal basis o... | Proof. We prove only (1), the proof of (2) being similar. Since \( f \) is a rotation and \( n = {2m} + 1 \) is odd, by Theorem 27.1, \( f \) is the composition of an even number less than or equal to \( {2m} \) of reflections. From Lemma 24.15, recall the Grassmann relation\n\n\[ \dim \left( M\right) + \dim \left( N\r... | Yes |
Theorem 27.5. Let \( E \) be a Euclidean space of dimension \( n \geq 3 \). Every rotation \( f \in \mathbf{{SO}}\left( E\right) \) is the composition of an even number of flips \( f = {f}_{2k} \circ \cdots \circ {f}_{1} \), where \( {2k} \leq n \). Furthermore, if \( u \neq 0 \) is invariant under \( f \) (i.e., \( u ... | Proof. By Theorem 27.1, the rotation \( f \) can be expressed as an even number of hyperplane reflections \( f = {s}_{2k} \circ {s}_{{2k} - 1} \circ \cdots \circ {s}_{2} \circ {s}_{1} \), with \( {2k} \leq n \). By Lemma 27.4, every composition of two reflections \( {s}_{2i} \circ {s}_{{2i} - 1} \) can be replaced by t... | Yes |
Proposition 27.7. Given any two nontrivial Euclidean affine spaces \( E \) and \( F \) of the same finite dimension \( n \), for every function \( f : E \rightarrow F \), the following properties are equivalent:\n\n(1) \( f \) is an affine map and \( \parallel \overrightarrow{f\left( a\right) f\left( b\right) }\paralle... | Proof. Obviously, (1) implies (2). In order to prove that (2) implies (1), we proceed as follows. First, we pick some arbitrary point \( \Omega \in E \) . We define the map \( g : \overrightarrow{E} \rightarrow \overrightarrow{F} \) such\n\nthat\n\[ g\left( u\right) = \overrightarrow{f\left( \Omega \right) f\left( {\Om... | Yes |
Let \( E \) be any affine space of finite dimension. For every affine map \( f : E \rightarrow E \), let \( \operatorname{Fix}\left( f\right) = \{ a \in E \mid f\left( a\right) = a\} \) be the set of fixed points of \( f \). The following properties hold:\n\n(1) If \( f \) has some fixed point \( a \), so that \( \oper... | Proof. (1) Since the identity\n\n\[ \overrightarrow{{\Omega f}\left( b\right) } - \overrightarrow{\Omega b} = \overrightarrow{{\Omega f}\left( \Omega \right) } + \overrightarrow{f}\left( \overrightarrow{\Omega b}\right) - \overrightarrow{\Omega b} \]\n\nholds for all \( \Omega, b \in E \), if \( f\left( a\right) = a \)... | Yes |
Proposition 27.9. Given any affine space \( E \), if \( f : E \rightarrow E \) and \( g : E \rightarrow E \) are affine orthogonal symmetries about parallel affine subspaces \( {F}_{1} \) and \( {F}_{2} \), then \( g \circ f \) is a translation defined by the vector \( 2\overrightarrow{ab} \), where \( \overrightarrow{... | Proof. We observed earlier that the linear maps \( \overrightarrow{f} \) and \( \overrightarrow{g} \) associated with \( f \) and \( g \) are the linear reflections about the directions of \( {F}_{1} \) and \( {F}_{2} \). However, \( {F}_{1} \) and \( {F}_{2} \) have the same direction, and so \( \overrightarrow{f} = \... | No |
Theorem 27.10. Let \( E \) be a Euclidean affine space of finite dimension \( n \) . For every affine isometry \( f : E \rightarrow E \), there is a unique affine isometry \( g : E \rightarrow E \) and a unique translation \( t = {t}_{\tau } \), with \( \overrightarrow{f}\left( \tau \right) = \tau \) (i.e., \( \tau \in... | Proof. The proof rests on the following two key facts:\n\n(1) If we can find some \( x \in E \) such that \( \overrightarrow{{xf}\left( x\right) } = \tau \) belongs to \( \operatorname{Ker}\left( {\overrightarrow{f} - \mathrm{{id}}}\right) \), we get the existence of \( g \) and \( \tau \) .\n\n(2) \( \overrightarrow{E... | Yes |
Let \( E \) be an affine Euclidean space of dimension \( n \geq 1 \). Every affine isometry \( f \in \mathbf{{Is}}\left( E\right) \) that has a fixed point and is not the identity is the composition of at most \( n \) affine reflections. Every affine isometry \( f \in \mathbf{{Is}}\left( E\right) \) that has no fixed p... | Proof. First, we use Theorem 27.10. If \( f \) has a fixed point \( \Omega \), we choose \( \Omega \) as an origin and work in the vector space \( {E}_{\Omega } \). Since \( f \) behaves as a linear isometry, the result follows from Theorem 27.1. More specifically, we can write \( \overrightarrow{f} = \overrightarrow{{... | Yes |
Theorem 27.12. Let \( E \) be a Euclidean affine space of dimension \( n \geq 3 \) . Every affine rigid motion \( f \in \mathbf{{SE}}\left( E\right) \) is the composition of an even number of affine flips \( f = {f}_{2k} \circ \cdots \circ {f}_{1} \) , where \( {2k} \leq n \) . | Proof. As in the proof of Theorem 27.11, we distinguish between the two cases where \( f \) has some fixed point or not. If \( f \) has a fixed point \( \Omega \), we apply Theorem 27.5. More specifically, we can write \( \overrightarrow{f} = \overrightarrow{{f}_{2k}} \circ \cdots \circ \overrightarrow{{f}_{1}} \) for ... | Yes |
Proposition 28.1. Let \( E \) be any nontrivial Hermitian space.\n\n(1) For any two vectors \( u, v \in E \) such that \( u \neq v \) and \( \parallel u\parallel = \parallel v\parallel \), if \( u \cdot v = {e}^{i\theta }\left| {u \cdot v}\right| \), then the (usual) reflection \( s \) about the hyperplane orthogonal t... | Proof. (1) Consider the (usual) reflection about the hyperplane orthogonal to \( w = v - {e}^{-{i\theta }}u \) .\n\nWe have\n\[ s\left( u\right) = u - 2\frac{\left( u \cdot \left( v - {e}^{-{i\theta }}u\right) \right) }{{\begin{Vmatrix}v - {e}^{-{i\theta }}u\end{Vmatrix}}^{2}}\left( {v - {e}^{-{i\theta }}u}\right) .\n\... | Yes |
Proposition 28.3. Let \( E \) be a nontrivial Hermitian space. For any two distinct orthogonal vectors \( u, v \) such that \( \parallel u\parallel = \parallel v\parallel \), we have\n\n\[{\rho }_{v, - \theta } \circ {\rho }_{u,\theta } = {h}_{v - u} \circ {h}_{v - {e}^{-{i\theta }}u} = {h}_{u + v} \circ {h}_{u + {e}^{... | Proof. Since \( u \) and \( v \) are orthogonal, each one is in the hyperplane orthogonal to the other, and thus,\n\n\[{\rho }_{u,\theta }\left( u\right) = {e}^{i\theta }u\]\n\n\[{\rho }_{u,\theta }\left( v\right) = v\]\n\n\[{\rho }_{v, - \theta }\left( u\right) = u\]\n\n\[{\rho }_{v, - \theta }\left( v\right) = {e}^{-... | Yes |
Proposition 28.4. Let \( E \) be a nontrivial Hermitian space, and let \( \left( {{u}_{1},\ldots ,{u}_{n}}\right) \) be some orthonormal basis for \( E \) . For any \( {\theta }_{1},\ldots ,{\theta }_{n} \) such that \( {\theta }_{1} + \cdots + {\theta }_{n} = 0 \), if \( f \in \mathbf{U}\left( n\right) \) is the isome... | Proof. It is obvious from the definitions that\n\n\[ f = {\rho }_{{u}_{n},{\theta }_{n}} \circ \cdots \circ {\rho }_{{u}_{1},{\theta }_{1}} \]\n\nand since the determinant of \( f \) is\n\n\[ D\left( f\right) = {e}^{i{\theta }_{1}}\cdots {e}^{i{\theta }_{n}} = {e}^{i\left( {{\theta }_{1} + \cdots + {\theta }_{n}}\right... | Yes |
Let \( E \) be a Hermitian space of dimension \( n \geq 1 \). Every rotation \( f \in \mathbf{{SU}}\left( E\right) \) different from the identity is the composition of at most \( {2n} - 2 \) standard hyperplane reflections. Every isometry \( f \in \mathbf{U}\left( E\right) \) different from the identity is the composit... | Proof. By Theorem 28.2, \( f \in \mathbf{{SU}}\left( n\right) \) can be written as a composition \[ {\rho }_{{u}_{n},{\theta }_{n}} \circ \cdots \circ {\rho }_{{u}_{1},{\theta }_{1}} \] where \( \left( {{u}_{1},\ldots ,{u}_{n}}\right) \) is an orthonormal basis of eigenvectors. Since \( f \) is a rotation, \( \det \lef... | Yes |
Theorem 28.6. Let \( E \) be a Hermitan space of dimension \( n \geq 3 \). Every rotation \( f \in \mathbf{{SU}}\left( E\right) \) is the composition of an even number of flips \( f = {f}_{2k} \circ \cdots \circ {f}_{1} \), where \( k \leq n - 1 \). Furthermore, if \( u \neq 0 \) is invariant under \( f \) (i.e. \( u \... | Proof. It is identical to that of Theorem 27.5, except that it uses Theorem 28.5 instead of Theorem 27.1. The second part of the Proposition also holds, because if \( u \neq 0 \) is an eigenvector of \( f \) for 1, then \( u \) is one of the vectors in the orthonormal basis of eigenvectors used in 28.2. The details are... | No |
Let \( E \) be a nontrivial Hermitian space of dimension \( n \) . Given any orthonormal basis \( \left( {{e}_{1},\ldots ,{e}_{n}}\right) \), for any \( n \) -tuple of vectors \( \left( {{v}_{1},\ldots ,{v}_{n}}\right) \), there is a sequence of \( n - 1 \) isometries \( {h}_{1},\ldots ,{h}_{n - 1} \), such that \( {h}... | The proof is very similar to the proof of Proposition 13.3, but it needs to be modified a little bit since Proposition 28.1 is weaker than Proposition 13.2. We explain how to modify the induction step, leaving the base case and the rest of the proof as an exercise.\n\nAs in the proof of Proposition 13.3, the vectors \(... | No |
Proposition 28.8. For every complex \( n \times n \) -matrix \( A \), there is a sequence \( {H}_{1},\ldots ,{H}_{n - 1} \) of matrices, where each \( {H}_{i} \) is either a Householder matrix or the identity, and an upper triangular matrix \( R \), such that\n\n\[ R = {H}_{n - 1}\cdots {H}_{2}{H}_{1}A \] | Proof. It is essentially identical to the proof of Proposition 13.4, and we leave the details as an exercise. For the last statement, observe that \( {h}_{n} \circ \cdots \circ {h}_{1} \) is also an isometry. | No |
Proposition 28.10. Given any two nontrivial Hermitian affine spaces \( E \) and \( F \) of the same finite dimension \( n \), for every function \( f : E \rightarrow F \), the following properties are equivalent:\n\n(1) \( f \) is an affine map and \( \parallel \overrightarrow{f\left( a\right) f\left( b\right) }\parall... | Proof. Obviously, (1) implies (2). The proof that that (2) implies (1) is similar to the proof of Proposition 27.7, but uses Proposition 14.14 instead of Proposition 12.12. The details are left as an exercise. | No |
Proposition 28.12. Given any affine complex space \( E \), if \( f : E \rightarrow E \) and \( g : E \rightarrow E \) are affine orthogonal symmetries about parallel affine subspaces \( {F}_{1} \) and \( {F}_{2} \), then \( g \circ f \) is a translation defined by the vector \( 2\overrightarrow{ab} \), where \( \overri... | It is easy to check that the proof of Proposition 27.10 also holds in the Hermitian case. | No |
Proposition 28.13. Let \( E \) be a Hermitian affine space of finite dimension \( n \) . For every affine isometry \( f : E \rightarrow E \), there is a unique affine isometry \( g : E \rightarrow E \) and a unique translation \( t = {t}_{\tau } \), with \( \overrightarrow{f}\left( \tau \right) = \tau \) (i.e., \( \tau... | \[ \overrightarrow{G} = \operatorname{Ker}\left( {\overrightarrow{f} - \mathrm{{id}}}\right) = E\left( {1,\overrightarrow{f}}\right) \] and such that \[ f = t \circ g\;\text{ and }\;t \circ g = g \circ t. \] Furthermore, we have the following additional properties: (a) \( f = g \) and \( \tau = 0 \) iff \( f \) has som... | Yes |
Theorem 28.14. Let \( E \) be an affine Hermitian space of dimension \( n \geq 1 \). Every affine isometry in \( \mathbf{{Is}}\left( {n,\mathbb{C}}\right) \) can be written as the composition of at most \( {2n} - 1 \) affine isometries if it has a fixed point, or else as the composition of at most \( {2n} + 1 \) affine... | Proof. The proof is very similar to the proof of Theorem 27.11, except that it uses Theorem 28.5 instead of Theorem 27.1. The details are left as an exercise. | No |
Theorem 28.15. Let \( E \) be a Hermitian affine space of dimension \( n \geq 3 \) . Every rigid motion \( f \in \mathbf{{SE}}\left( {E,\mathbb{C}}\right) \) is the composition of an even number of affine flips \( f = {f}_{2k} \circ \cdots \circ {f}_{1} \), where \( k \leq n - 1 \) . | Proof. It is very similar to the proof of theorem 27.12, but it uses Proposition 28.6 instead of Proposition 27.5. The details are left as an exercise. | No |
Proposition 29.1. Given a bilinear map \( \varphi : E \times F \rightarrow K \), the following properties hold:\n\n(a) The map \( {l}_{\varphi } \) is injective iff Property (1) of Definition 29.5 holds.\n\n(b) The map \( {r}_{\varphi } \) is injective iff Property (2) of Definition 29.5 holds.\n\n(c) The bilinear form... | Proof. (a) Assume that (1) of Definition 29.5 holds. If \( {l}_{\varphi }\left( u\right) = 0 \), then \( {l}_{\varphi }\left( u\right) \) is the linear form whose value is 0 for all \( y \) ; that is,\n\n\[ \n{l}_{\varphi }\left( u\right) \left( y\right) = \varphi \left( {u, y}\right) = 0\;\text{ for all }y \in F, \n\]... | Yes |
Proposition 29.3. Given a bilinear map \( \varphi : E \times F \rightarrow K \), if \( \varphi \) is nondegenerate and \( E \) and \( F \) are finite-dimensional, then \( \dim \left( E\right) = \dim \left( F\right) = n \), and for every basis \( \left( {{e}_{1},\ldots ,{e}_{n}}\right) \) of \( E \) , there is a basis \... | Proof. Since \( \varphi \) is nondegenerate, by Proposition 29.1 we have \( \dim \left( E\right) = \dim \left( F\right) = n \), and by Proposition 29.2, the linear map \( {r}_{\varphi } \) is bijective. Then, if \( \left( {{e}_{1}^{ * },\ldots ,{e}_{n}^{ * }}\right) \) is the dual basis (in \( \left. {E}^{ * }\right) \... | Yes |
Theorem 29.4. Given any bilinear form \( \varphi : E \times E \rightarrow K \) with \( \dim \left( E\right) = n \), if \( \varphi \) is symmetric (possibly degenerate) and \( K \) does not have characteristic 2, then there is a basis \( \left( {{e}_{1},\ldots ,{e}_{n}}\right) \) of \( E \) such that \( \varphi \left( {... | Proof. We proceed by induction on \( n \geq 0 \), following a proof due to Chevalley. The base case \( n = 0 \) is trivial. For the induction step, assume that \( n \geq 1 \) and that the induction hypothesis holds for all vector spaces of dimension \( n - 1 \) . If \( \varphi \left( {u, v}\right) = 0 \) for all \( u, ... | Yes |
Proposition 29.6. Given any bilinear form \( \varphi : E \times E \rightarrow K \) with \( \dim \left( E\right) = n \), if \( \varphi \) is symmetric and \( K \) does not have characteristic 2, then there is a basis \( \left( {{e}_{1},\ldots ,{e}_{n}}\right) \) of \( E \) such that | Proof. The first statement is a direct consequence of Theorem 29.4. If \( K = \mathbb{C} \), then every \( {\lambda }_{i} \) has a square root \( {\mu }_{i} \), and if replace \( {e}_{i} \) by \( {e}_{i}/{\mu }_{i} \), we obtained the desired form.\n\nIf \( K = \mathbb{R} \), then there are two cases:\n\n1. If \( {\lam... | Yes |
Proposition 29.8. If \( \varphi \) is a nonzero Hermitian or skew-Hermitian form and if \( \varphi \left( {u, u}\right) = 0 \) for all \( u \in E \), then \( K \) is of characteristic 2 and the automorphism \( \lambda \mapsto \bar{\lambda } \) is the identity. | Proof. We give the proof in the Hermitian case, the skew-Hermitian case being left as an exercise. Assume that \( \varphi \) is alternating. From the identity\n\n\[ \varphi \left( {u + v, u + v}\right) = \varphi \left( {u, u}\right) + \varphi \left( {u, v}\right) + \overline{\varphi \left( {u, v}\right) } + \varphi \le... | No |
Proposition 29.11. Given any Hermitian form \( \varphi : E \times E \rightarrow \mathbb{C} \) with \( \dim \left( E\right) = n \), there is a basis \( \left( {{e}_{1},\ldots ,{e}_{n}}\right) \) of \( E \) such that\n\n\[ \Phi \left( {\mathop{\sum }\limits_{{i = 1}}^{n}{x}_{i}{e}_{i}}\right) = \mathop{\sum }\limits_{{i ... | The proof of Proposition 29.11 is the same as the real case of Proposition 29.6. Sylvester's inertia law (Proposition 29.7) also holds for Hermitian forms: \( p \) and \( q \) only depend on \( \varphi \) . | No |
Proposition 29.12. For any sesquilinear form \( \varphi : E \times F \rightarrow K \), the space \( E/{F}^{ \bot } \) is finite-dimensional iff the space \( F/{E}^{ \bot } \) is finite-dimensional; if so, \( \dim \left( {E/{F}^{ \bot }}\right) = \dim \left( {F/{E}^{ \bot }}\right) \) . | Proof. Since the sesquilinear form \( \left\lbrack \varphi \right\rbrack : \left( {E/{F}^{ \bot }}\right) \times \left( {F/{E}^{ \bot }}\right) \rightarrow K \) is nondegenerate, the maps \( {l}_{\left\lbrack \varphi \right\rbrack } : \overline{\left( E/{F}^{ \bot }\right) } \rightarrow {\left( F/{E}^{ \bot }\right) }^... | Yes |
Proposition 29.13. Let \( \varphi : E \times F \rightarrow K \) be any nondegenerate sesquilinear form. A subspace \( U \) of \( E \) has finite dimension iff \( {U}^{ \bot } \) has finite codimension in \( F \) . If \( \dim \left( U\right) \) is finite, then \( \operatorname{codim}\left( {U}^{ \bot }\right) = \dim \le... | Proof. Since \( \varphi \) is nondegenerate \( {E}^{ \bot } = \{ 0\} \) and \( {F}^{ \bot } = \{ 0\} \), so Proposition 29.12 applied to the restriction of \( \varphi \) to \( U \times F \) implies that a subspace \( U \) of \( E \) has finite dimension iff \( {U}^{ \bot } \) has finite codimension in \( F \), and that... | Yes |
Let \( \varphi : E \times F \rightarrow K \) be any sesquilinear form. Given any two subspaces \( U \) and \( V \) of \( E \), we have\n\n\[{\left( U + V\right) }^{ \bot } = {U}^{ \bot } \cap {V}^{ \bot }\] | Proof. If \( w \in {\left( U + V\right) }^{ \bot } \), then \( \varphi \left( {u + v, w}\right) = 0 \) for all \( u \in U \) and all \( v \in V \) . In particular, with \( v = 0 \), we have \( \varphi \left( {u, w}\right) = 0 \) for all \( u \in U \), and with \( u = 0 \), we have \( \varphi \left( {v, w}\right) = 0 \)... | Yes |
Proposition 29.15. Let \( \varphi : E \times F \rightarrow K \) be any sesquilinear form. If \( \varphi \) has finite rank \( r \) , then \( {l}_{\varphi } \) and \( {r}_{\varphi } \) have the same rank, which is equal to \( r \) . | Proof. Because for every \( u \in E \) ,\n\n\[ \n{l}_{\varphi }\left( u\right) \left( y\right) = \overline{\varphi \left( {u, y}\right) }\;\text{ for all }y \in F, \n\] \n\nand for every \( v \in F \) ,\n\n\[ \n{r}_{\varphi }\left( v\right) \left( x\right) = \varphi \left( {x, v}\right) \;\text{ for all }x \in E, \n\] ... | Yes |
Proposition 29.16. With the same assumptions as in Definition 29.14 (which imply that \( {\varphi }_{1} \) is nondegenerate), if \( f : {E}_{1} \rightarrow {E}_{2} \) is a bijective linear map, then we have\n\n\[ \n{\varphi }_{1}\left( {x, y}\right) = {\varphi }_{2}\left( {f\left( x\right), f\left( y\right) }\right) \;... | Proof. We have\n\n\[ \n{\varphi }_{1}\left( {x, y}\right) = {\varphi }_{2}\left( {f\left( x\right), f\left( y\right) }\right) \n\]\n\niff\n\n\[ \n{\varphi }_{1}\left( {x, y}\right) = {\varphi }_{2}\left( {f\left( x\right), f\left( y\right) }\right) = {\varphi }_{1}\left( {x,{f}^{{ * }_{l}}\left( {f\left( y\right) }\rig... | Yes |
Proposition 29.18. (1) If \( \varphi : E \times E \rightarrow K \) is a sesquilinear map and if \( {l}_{\varphi } \) is injective, then for every linear map \( f : E \rightarrow E \), if\n\n\[ \varphi \left( {f\left( x\right), f\left( y\right) }\right) = \varphi \left( {x, y}\right) \;\text{ for all }x, y \in E, \]\n\n... | Proof. (1) If \( f\left( x\right) = 0 \), then\n\n\[ \varphi \left( {x, y}\right) = \varphi \left( {f\left( x\right), f\left( y\right) }\right) = \varphi \left( {0, f\left( y\right) }\right) = 0\;\text{ for all }y \in E. \]\n\nSince \( {l}_{\varphi } \) is injective, we must have \( x = 0 \), and thus \( f \) is inject... | Yes |
Proposition 29.19. Given an \( \epsilon \) -Hermitian form \( \varphi : E \times E \rightarrow K \) on \( E \), we have:\n\n(a) If \( U \) and \( V \) are any two orthogonal subspaces of \( E \), then\n\n\[ \operatorname{rad}\left( {U + V}\right) = \operatorname{rad}\left( U\right) + \operatorname{rad}\left( V\right) \... | Proof. (a) If \( U \) and \( V \) are orthogonal, then \( U \subseteq {V}^{ \bot } \) and \( V \subseteq {U}^{ \bot } \) . We get\n\n\[ \operatorname{rad}\left( {U + V}\right) = \left( {U + V}\right) \cap {\left( U + V\right) }^{ \bot } \]\n\n\[ = \left( {U + V}\right) \cap {U}^{ \bot } \cap {V}^{ \bot } \]\n\n\[ = U \... | Yes |
Proposition 29.20. Given an \( \epsilon \) -Hermitian form \( \varphi : E \times E \rightarrow K \) on \( E \), if \( U \) is a finite-dimensional nondegenerate subspace of \( E \), then \( E = U \oplus {U}^{ \bot } \) . | Proof. By hypothesis, the restriction \( {\varphi }_{U} \) of \( \varphi \) to \( U \) is nondegenerate, so the semilinear map \( {r}_{{\varphi }_{U}} : U \rightarrow {U}^{ * } \) is injective. Since \( U \) is finite-dimensional, \( {r}_{{\varphi }_{U}} \) is actually bijective, so for every \( v \in E \), if we consi... | Yes |
Proposition 29.21. Given an \( \epsilon \) -Hermitian form \( \varphi : E \times E \rightarrow K \) on \( E \), if \( \varphi \) is nondegenerate and if \( U \) is a finite-dimensional subspace of \( E \), then \( \operatorname{rad}\left( U\right) = \operatorname{rad}\left( {U}^{ \bot }\right) \), and the following con... | Proof. By definition, \( \operatorname{rad}\left( {U}^{ \bot }\right) = {U}^{ \bot } \cap {U}^{ \bot \bot } \), and since \( \varphi \) is nondegenerate and \( U \) is finite-dimensional, \( {U}^{ \bot \bot } = U \), so \( \operatorname{rad}\left( {U}^{ \bot }\right) = {U}^{ \bot } \cap {U}^{ \bot \bot } = U \cap {U}^{... | Yes |
Proposition 29.22. Given an \( \epsilon \) -Hermitian form \( \varphi : E \times E \rightarrow K \) on a finite-dimensional space \( E \), if \( \varphi \) is nondegenerate, then for every subspace \( U \) of \( E \) we have\n\n(i) \( \dim \left( U\right) + \dim \left( {U}^{ \bot }\right) = \dim \left( E\right) \).\n\n... | Proof. (i) Since \( \varphi \) is nondegenerate and \( E \) is finite-dimensional, the semilinear map \( {l}_{\varphi } : E \rightarrow \) \( {E}^{ * } \) is bijective. By transposition, the inclusion \( i : U \rightarrow E \) yields a surjection \( r : {E}^{ * } \rightarrow {U}^{ * } \) (with \( r\left( f\right) = f \... | Yes |
Proposition 29.23. Let \( \varphi : E \times E \rightarrow K \) be an alternating bilinear form on \( E \) . If \( u, v \in E \) are two (nonzero) vectors such that \( \varphi \left( {u, v}\right) = \lambda \neq 0 \), then \( u \) and \( v \) are linearly independent. If we let \( {u}_{1} = {\lambda }^{-1}u \) and \( {... | Proof. If \( u \) and \( v \) were linearly dependent, as \( u, v \neq 0 \), we could write \( v = {\mu u} \) for some \( \mu \neq 0 \) , but then, since \( \varphi \) is alternating, we would have\n\n\[ \lambda = \varphi \left( {u, v}\right) = \varphi \left( {u,{\mu u}}\right) = {\mu \varphi }\left( {u, u}\right) = 0,... | Yes |
Theorem 29.24. Let \( \varphi : E \times E \rightarrow K \) be an alternating bilinear form on a space \( E \) of finite dimension \( n \) . Then, there is a direct sum decomposition of \( E \) into pairwise orthogonal subspaces\n\n\[ E = {W}_{1} \oplus \cdots \oplus {W}_{r} \oplus \operatorname{rad}\left( E\right) \]\... | Proof. If \( \varphi = 0 \), then \( E = {E}^{ \bot } \) and we are done. Otherwise, there are two nonzero vectors \( u, v \in E \) such that \( \varphi \left( {u, v}\right) \neq 0 \), so by Proposition 29.23, we obtain a hyperbolic plane \( {W}_{2} \) spanned by two vectors \( {u}_{1},{v}_{1} \) such that \( \varphi \... | Yes |
Proposition 29.25. Let \( \varphi : E \times E \rightarrow K \) be an alternating bilinear form on a space \( E \) of finite dimension \( n \) .\n\n(1) The rank of \( \varphi \) is even.\n\n(2) If \( \varphi \) is nondegenerate, then \( \dim \left( E\right) = n \) is even.\n\n(3) Two alternating bilinear forms \( {\var... | The only part that requires a proof is part (3), which is left as an easy exercise. | No |
Proposition 29.26. Let \( \varphi : E \times E \rightarrow K \) be a nondegenerate symmetric bilinear form with \( K \) a field of characteristic different from 2. For any nonzero isotropic vector \( u \), there is another nonzero isotropic vector \( v \) such that \( \varphi \left( {u, v}\right) = 2 \), and \( u \) an... | Proof. Since \( \varphi \) is nondegenerate, there is some nonzero vector \( z \) such that (rescaling \( z \) if necessary) \( \varphi \left( {u, z}\right) = 1 \) . If\n\n\[ v = {2z} - \varphi \left( {z, z}\right) u \]\n\nthen since \( \varphi \left( {u, u}\right) = 0 \) and \( \varphi \left( {u, z}\right) = 1 \), not... | Yes |
Proposition 29.27. If \( \Phi \) is any nondegenerate quadratic form (over a field of characteristic \( \neq 2) \) such that there is some nonzero vector \( x \in E \) with \( \Phi \left( x\right) = 0 \), then for every \( \alpha \in K \) , there is some \( y \in E \) such that \( \Phi \left( y\right) = \alpha \) . | Proof. Since by hypothesis there is some nonzero vector \( u \in E \) with \( \Phi \left( u\right) = 0 \), by Proposition 29.26 there is another isotropic vector \( v \) such that \( u \) and \( v \) are linearly independent and such that (after rescaling) \( \varphi \left( {u, v}\right) = 1 \) . Then for any \( \alpha... | Yes |
Lemma 29.28. Let \( \varphi \) be an \( \epsilon \) -Hermitian form on \( E \) and assume that \( \varphi \) satisfies property (T). For any totally isotropic subspace \( U \neq \left( 0\right) \) of \( E \), for every \( x \in E \) not orthogonal to \( U \) , and for every \( \alpha \in K \), there is some \( y \in U ... | Proof. By property (T), we have \( \varphi \left( {x, x}\right) = \beta + \epsilon \bar{\beta } \) for some \( \beta \in K \) . For any \( y \in U \), since \( \varphi \) is \( \epsilon \) -Hermitian, \( \varphi \left( {y, x}\right) = {\epsilon \varphi }\left( {x, y}\right) \), and since \( U \) is totally isotropic \(... | Yes |
Proposition 29.29. Let \( \varphi \) be an \( \epsilon \) -Hermitian form on \( E \), assume that \( \varphi \) is nondegenerate and satisfies property (T), and let \( U \) be any totally isotropic subspace of \( E \) of finite dimension \( \dim \left( U\right) = r \geq 1 \n\n(1) If \( {U}^{\prime } \) is any totally i... | Proof. (1) Let \( {\varphi }^{\prime } \) be the restriction of \( \varphi \) to \( U \times {U}^{\prime } \) . Since \( {U}^{\prime } \cap {U}^{ \bot } = \left( 0\right) \), for any \( v \in {U}^{\prime } \) , if \( \varphi \left( {u, v}\right) = 0 \) for all \( u \in U \), then \( v = 0 \) . Thus, \( {\varphi }^{\pri... | Yes |
Proposition 29.32. Let \( \varphi \) be an \( \epsilon \) -Hermitian form on \( E \) which is nondegenerate and satisfies property (T). For any subspace \( U \) of \( E \) of finite dimension, if we write\n\n\[ U = V\overset{ \bot }{ \oplus }W \]\n\nfor some orthogonal complement \( W \) of \( V = \operatorname{rad}\le... | Proof. Since \( W \) is nondegenerate, \( {W}^{ \bot } \) is also nondegenerate, and \( V \subseteq {W}^{ \bot } \) . Therefore, we can apply Theorem 29.30 to the restriction of \( \varphi \) to \( {W}^{ \bot } \) and to \( V \) to obtain the required \( {V}^{\prime } \) . We know that \( V \oplus {V}^{\prime } \) is n... | No |
Theorem 29.33. Let \( \varphi \) be an \( \epsilon \) -Hermitian form on \( E \) which is nondegenerate and satisfies property (T). Let \( {U}_{1} \) and \( {U}_{2} \) be two totally isotropic maximal subspaces of \( E \), with \( {U}_{1} \) or \( {U}_{2} \) of finite dimension \( \geq 1 \) . Write \( U = {U}_{1} \cap ... | Proof. First observe that if \( X \) is a totally isotropic maximal subspace of \( E \), then any isotropic vector \( x \in E \) orthogonal to \( X \) must belong to \( X \), since otherwise, \( X + {Kx} \) would be a totally isotropic subspace strictly containing \( X \), contradicting the maximality of \( X \) . As a... | Yes |
Theorem 29.34. Let \( \varphi \) be an \( \epsilon \) -Hermitian form on \( E \) which is nondegenerate and satisfies property (T).\n\n(1) Any two totally isotropic maximal spaces of finite dimension have the same dimension.\n\n(2) For any totally isotropic maximal subspace \( U \) of finite dimension \( r \geq 1 \), t... | Proof. Part (1) follows from Theorem 29.33. By Proposition 29.30, we obtain a totally isotropic subspace \( {U}^{\prime } \) of dimension \( r \) such that \( U \cap {U}^{\prime } = \left( 0\right) \) . By applying Theorem 29.33 to \( {U}_{1} = U \) and \( {U}_{2} = {U}^{\prime } \), we get \( U = W = \left( 0\right) \... | Yes |
Proposition 29.35. If \( K \) is algebraically closed and \( E \) has dimension \( n \), then for every nondegenerate quadratic form \( \Phi \), there is a basis \( \left( {{e}_{1},\ldots ,{e}_{n}}\right) \) such that \( \Phi \) is given by\n\n\[ \Phi \left( {\mathop{\sum }\limits_{{i - 1}}^{n}{x}_{i}{e}_{i}}\right) = ... | Proof. We work with the polar form \( \varphi \) of \( \Phi \) . Let \( {U}_{1} \) and \( {U}_{2} \) be some totally isotropic subspaces such that \( {U}_{1} \cap {U}_{2} = \left( 0\right) \) given by Theorem 29.34, and let \( q \) be their common dimension. Then, \( W = U = \left( 0\right) \) . Since we can pick bases... | Yes |
Proposition 29.36. Given any two nonzero vectors \( u, v \in E \), there is a symplectic map \( f \) such that \( f\left( u\right) = v \), and \( f \) is either a symplectic transvection, or the composition of two symplectic transvections. | Proof. There are two cases.\n\nCase 1. \( \varphi \left( {u, v}\right) \neq 0 \) .\n\nIn this case, \( u \neq v \), since \( \varphi \left( {u, u}\right) = 0 \) . Let us look for a symplectic transvection of the form \( {\tau }_{v - u,\lambda } \) . We want\n\n\[ v = u + {\lambda \varphi }\left( {v - u, u}\right) \left... | Yes |
Given any two hyperbolic planes \( {W}_{1} \) and \( {W}_{2} \) given by bases \( \left( {{u}_{1},{v}_{1}}\right) \) and \( \left( {{u}_{2},{v}_{2}}\right) \) (with \( \varphi \left( {{u}_{i},{u}_{i}}\right) = \varphi \left( {{v}_{i},{v}_{i}}\right) = 0 \) and \( \varphi \left( {{u}_{i},{v}_{i}}\right) = 1 \), for \( i... | Proof. From Proposition 29.36, we can map \( {u}_{1} \) to \( {u}_{2} \), using a map \( f \) which is the composition of at most two symplectic transvections. Say \( {v}_{3} = f\left( {v}_{1}\right) \) . We claim that there is a map \( g \) such that \( g\left( {u}_{2}\right) = {u}_{2} \) and \( g\left( {v}_{3}\right)... | Yes |
The symplectic group \( \mathbf{{Sp}}\left( {{2m}, K}\right) \) is generated by the symplectic transvec-tions. For every transvection \( f \in \mathbf{{Sp}}\left( {{2m}, K}\right) \), we have \( \det \left( f\right) = 1 \) . | Proof. Let \( G \) be the subgroup of \( \mathbf{{Sp}}\left( {{2m}, K}\right) \) generated by the transvections. We need to prove that \( G = \mathbf{{Sp}}\left( {{2m}, K}\right) \) . Let \( \left( {{u}_{1},{v}_{1},\ldots ,{u}_{m},{v}_{m}}\right) \) be a symplectic basis of \( E \), and let \( f \in \) \( \mathbf{{Sp}}... | Yes |
Proposition 29.39. Let \( \varphi \) be a nondegenerate symmetric bilinear form on a vector space \( E \) . For any two nonzero vectors \( u, v \in E \), if \( \varphi \left( {u, u}\right) = \varphi \left( {v, v}\right) \) and \( v - u \) is nonisotropic, then the hyperplane reflection \( {\tau }_{H} = {\tau }_{v - u} ... | Proof. Since \( v - u \) is not isotropic, \( \varphi \left( {v - u, v - u}\right) \neq 0 \), and we have\n\n\[ \n{\tau }_{v - u}\left( u\right) = u - 2\frac{\varphi \left( {u, v - u}\right) }{\varphi \left( {v - u, v - u}\right) }\left( {v - u}\right) \n\]\n\n\[ \n= u - 2\frac{\varphi \left( {u, v}\right) - \varphi \l... | Yes |
Proposition 29.42. Let \( \left( {E,\varphi }\right) \) be an Artinian space of dimension \( {2m} \), and let \( U \) be a totally isotropic subspace of dimension \( m \) . For any isometry \( f \in \mathbf{O}\left( \varphi \right) \), if \( f\left( U\right) = U \), then \( \det \left( f\right) = 1 \) ( \( f \) is a ro... | Proof. We know that we can find a basis \( \left( {{u}_{1},\ldots ,{u}_{m},{v}_{1},\ldots ,{v}_{m}}\right) \) of \( E \) such \( \left( {{u}_{1},\ldots ,{u}_{m}}\right) \) is a basis of \( U \) and \( \varphi \) is represented by the matrix\n\n\[ \left( \begin{matrix} 0 & {I}_{m} \\ {I}_{m} & 0 \end{matrix}\right) \]\n... | Yes |
Proposition 29.44. Given a finite-dimensional space \( E \) equipped with an \( \epsilon \) -Hermitan form \( \varphi \), if \( {U}_{1} \) and \( {U}_{2} \) are two subspaces of \( E \) such that \( {U}_{1} \cap {U}_{2} = \left( 0\right) \) and if we have metric linear maps \( {f}_{1} : {U}_{1} \rightarrow E \) and \( ... | Proof. Indeed, since \( {f}_{1} \) and \( {f}_{2} \) are metric and using \( \left( *\right) \), we have\n\n\[ \varphi \left( {{f}_{1}\left( {u}_{1}\right) + {f}_{2}\left( {u}_{2}\right) ,{f}_{1}\left( {v}_{1}\right) + {f}_{2}\left( {v}_{2}\right) }\right) = \varphi \left( {{f}_{1}\left( {u}_{1}\right) ,{f}_{1}\left( {... | Yes |
Theorem 29.45. (Witt,1936) Let \( E \) and \( {E}^{\prime } \) be two finite-dimensional spaces respectively equipped with two nondegenerate \( \epsilon \) -Hermitan forms \( \varphi \) and \( {\varphi }^{\prime } \) satisfying condition (T), and assume that there is an isometry between \( \left( {E,\varphi }\right) \)... | Proof. Since \( \left( {E,\varphi }\right) \) and \( \left( {{E}^{\prime },{\varphi }^{\prime }}\right) \) are isometric, we may assume that \( {E}^{\prime } = E \) and \( {\varphi }^{\prime } = \varphi \) (if \( h : E \rightarrow {E}^{\prime } \) is an isometry, then \( {h}^{-1} \circ f \) is an injective metric map f... | No |
Theorem 29.46. (Witt Cancellation Theorem) Let \( \left( {{E}_{1},{\varphi }_{1}}\right) \) and \( \left( {{E}_{2},{\varphi }_{2}}\right) \) be two pairs of finite-dimensional spaces and nondegenerate \( \epsilon \) -Hermitian forms satisfying condition (T), and assume that \( \left( {{E}_{1},{\varphi }_{1}}\right) \) ... | Proof. If \( f : U \rightarrow V \) is an isometry between \( U \) and \( V \), by Witt’s theorem (Theorem 29.46), the linear map \( f \) extends to an isometry \( g \) between \( {E}_{1} \) and \( {E}_{2} \) . We claim that \( g \) maps \( {U}^{ \bot } \) into \( {V}^{ \bot } \) . This is because if \( v \in {U}^{ \bo... | Yes |
Theorem 29.48. (Witt-Sharpened Version) Let \( E \) be a finite-dimensional space equipped with a nondegenerate symmetric bilinear forms \( \varphi \) . For any subspace \( U \) of \( E \), every linear injective metric map \( f \) from \( U \) into \( E \) extends to an isometry \( g \) of \( E \) with a prescribed va... | Proof. If \( {g}_{1} \) and \( {g}_{2} \) are two extensions of \( f \) such that \( \\det \\left( {g}_{1}\\right) \\det \\left( {g}_{2}\\right) = - 1 \), then \( h = {g}_{1}^{-1} \\circ {g}_{2} \) is an isometry such that \( \\det \\left( h\\right) = - 1 \), and \( h \) leaves every vector of \( U \) fixed. Conversely... | Yes |
Proposition 30.1. Given two nonnull polynomials \( P\left( X\right) = {a}_{0} + {a}_{1}X + \cdots + {a}_{m}{X}^{m} \) of degree \( m \) and \( Q\left( X\right) = {b}_{0} + {b}_{1}X + \cdots + {b}_{n}{X}^{n} \) of degree \( n \), if either \( {a}_{m} \) or \( {b}_{n} \) is not a zero divisor, then \( {a}_{m}{b}_{n} \neq... | Proof. Since the coefficient of \( {X}^{m + n} \) in \( {PQ} \) is \( {a}_{m}{b}_{n} \), and since we assumed that either \( {a}_{m} \) or \( {a}_{n} \) is not a zero divisor, we have \( {a}_{m}{b}_{n} \neq 0 \), and thus, \( {PQ} \neq 0 \) and \[ \deg \left( {PQ}\right) = \deg \left( P\right) + \deg \left( Q\right) \]... | Yes |
Proposition 30.4. Let \( A \) be a ring, let \( f\left( X\right), g\left( X\right) \in A\left\lbrack X\right\rbrack \) be two polynomials of degree \( m = \deg \left( f\right) \) and \( n = \deg \left( g\right) \) with \( f\left( X\right) \neq 0 \), and assume that the leading coefficient \( {a}_{m} \) of \( f\left( X\... | Proof. We first prove the existence of \( q \) and \( r \) . Let\n\n\[ f = {a}_{m}{X}^{m} + {a}_{m - 1}{X}^{m - 1} + \cdots + {a}_{0}, \]\n\nand\n\n\[ g = {b}_{n}{X}^{n} + {b}_{n - 1}{X}^{n - 1} + \cdots + {b}_{0}. \]\n\nIf \( n < m \), then let \( q = 0 \) and \( r = g \) . Since \( \deg \left( g\right) < \deg \left( ... | Yes |
Proposition 30.6. Given any ring homomorphism \( h : A \rightarrow B \), the kernel \( \operatorname{Ker}h = \{ a \in A \mid \) \( h\left( a\right) = 0\} \) of \( h \) is an ideal. | Proof. Given \( a, b \in A \), we have \( a, b \in \operatorname{Ker}h \) iff \( h\left( a\right) = h\left( b\right) = 0 \), and since \( h \) is a homomorphism, we get\n\n\[ h\left( {b - a}\right) = h\left( b\right) - h\left( a\right) = 0, \]\n\nand\n\n\[ h\left( {ax}\right) = h\left( a\right) h\left( x\right) = 0 \]\... | Yes |
Given any ring \( A \) and any ideal \( \mathfrak{I} \subseteq A \), the equivalence relation \( { \equiv }_{\mathfrak{I}} \) defined by \( a{ \equiv }_{\mathfrak{I}}b \) iff \( b - a \in \mathfrak{I} \) is a congruence, which means that if \( {a}_{1}{ \equiv }_{\mathfrak{I}}{b}_{1} \) and \( {a}_{2}{ \equiv }_{\mathfr... | Proof. Everything is straightforward. For example, if \( {a}_{1}{ \equiv }_{\mathfrak{I}}{b}_{1} \) and \( {a}_{2}{ \equiv }_{\mathfrak{I}}{b}_{2} \), then \( {b}_{1} - {a}_{1} \in \mathfrak{I} \) and \( {b}_{2} - {a}_{2} \in \mathfrak{I} \) . Since \( \mathfrak{I} \) is an ideal, we get\n\n\[ \left( {{b}_{1} - {a}_{1}... | Yes |
Given a ring \( A \), for any ideal \( \mathfrak{I} \subseteq A \), the following properties hold.\n\n(1) The ideal \( \mathfrak{I} \) is a prime ideal iff \( A/\mathfrak{I} \) is an integral domain. | Proof. (1) Assume that \( \mathfrak{I} \) is a prime ideal. Since \( \mathfrak{I} \) is prime, \( \mathfrak{I} \neq A \), and thus, \( A/\mathfrak{I} \) is not the trivial ring (0). If \( \left\lbrack a\right\rbrack \left\lbrack b\right\rbrack = 0 \), since \( \left\lbrack a\right\rbrack \left\lbrack b\right\rbrack = \... | Yes |
Corollary 30.9. Given any ring \( A \), every maximal ideal \( \mathfrak{I} \) in \( A \) is a prime ideal. | Proof. If \( \mathfrak{I} \) is a maximal ideal, then, by Proposition 30.8, the quotient ring \( A/\mathfrak{I} \) is a field. However, a field is an integral domain, and by Proposition 30.8 (again), \( \mathfrak{I} \) is a prime ideal. | Yes |
Proposition 30.10. Let \( K \) be a field. The following properties hold:\n\n(1) For any two nonzero polynomials \( f, g \in K\left\lbrack X\right\rbrack ,\left( f\right) = \left( g\right) \) iff there is some \( \lambda \neq 0 \) in \( K \) such that \( g = {\lambda f} \) . | Proof. (1) If \( \left( f\right) = \left( g\right) \), there are some nonzero polynomials \( {q}_{1},{q}_{2} \in K\left\lbrack X\right\rbrack \) such that \( g = f{q}_{1} \) and \( f = g{q}_{2} \) . Thus, we have \( f = f{q}_{1}{q}_{2} \), which implies \( f\left( {1 - {q}_{1}{q}_{2}}\right) = 0 \) . Since \( K \) is a... | Yes |
Proposition 30.11. Let \( K \) be a field and let \( f, g \in K\left\lbrack X\right\rbrack \) be any two nonzero polynomials. For every polynomial \( d \in K\left\lbrack X\right\rbrack \), the following properties are equivalent:\n\n(1) The polynomial \( d \) is a gcd of \( f \) and \( g \) .\n\n(2) The polynomial \( d... | Proof. Given any two nonzero polynomials \( u, v \in K\left\lbrack X\right\rbrack \), observe that \( u \) divides \( v \) iff \( \left( v\right) \subseteq \left( u\right) \) . Now,(2) can be restated as \( \left( f\right) \subseteq \left( d\right) ,\left( g\right) \subseteq \left( d\right) \), and \( d \in \left( f\ri... | Yes |
Proposition 30.12. Let \( K \) be a field and let \( f, g \in K\left\lbrack X\right\rbrack \) be any two nonzero polynomials. For every \( \gcd d \in K\left\lbrack X\right\rbrack \) of \( f \) and \( g \), the following properties hold:\n\n(1) For every nonzero polynomial \( q \in K\left\lbrack X\right\rbrack \), the p... | Proof. (1) By Proposition 30.11 (2), \( d \) divides \( f \) and \( g \), and there exist \( u, v \in K\left\lbrack X\right\rbrack \), such that\n\n\[ d = {uf} + {vg}. \]\n\nThen, \( {dq} \) divides \( {fq} \) and \( {gq} \), and\n\n\[ {dq} = {ufq} + {vgq}. \]\n\nBy Proposition 30.11 (2), \( {dq} \) is a gcd of \( {fq}... | Yes |
Proposition 30.13. (Euclid’s proposition) Let \( K \) be a field and let \( f, g, h \in K\left\lbrack X\right\rbrack \) be any nonzero polynomials. If \( f \) divides \( {gh} \) and \( f \) is relatively prime to \( g \), then \( f \) divides \( h \) . | Proof. From Proposition 30.11, \( f \) and \( g \) are relatively prime iff there exist some polynomials \( u, v \in K\left\lbrack X\right\rbrack \) such that\n\n\[
{uf} + {vg} = 1\text{.}\]\n\nThen, we have\n\n\[
{ufh} + {vgh} = h,\]\n\nand since \( f \) divides \( {gh} \), it divides both \( {ufh} \) and \( {vgh} \),... | Yes |
Proposition 30.14. Let \( K \) be a field and let \( f,{g}_{1},\ldots ,{g}_{m} \in K\left\lbrack X\right\rbrack \) be some nonzero polynomials. If \( f \) and \( {g}_{i} \) are relatively prime for all \( i,1 \leq i \leq m \), then \( f \) and \( {g}_{1}\cdots {g}_{m} \) are relatively prime. | Proof. We proceed by induction on \( m \) . The case \( m = 1 \) is trivial. Let \( h = {g}_{2}\cdots {g}_{m} \) . By the induction hypothesis, \( f \) and \( h \) are relatively prime. Let \( d \) be a gcd of \( f \) and \( {g}_{1}h \) . We claim that \( d \) is relatively prime to \( {g}_{1} \) . Otherwise, \( d \) a... | Yes |
Proposition 30.15. Let \( K \) be a field and let \( {f}_{1},\ldots ,{f}_{n} \in K\left\lbrack X\right\rbrack \) be any \( n \geq 2 \) nonzero polynomials. For every polynomial \( d \in K\left\lbrack X\right\rbrack \), the following properties are equivalent:\n\n(1) The polynomial \( d \) is a gcd of \( {f}_{1},\ldots ... | In addition, \( d \neq 0 \), and \( d \) is unique up to multiplication by a nonzero scalar in \( K \) . | No |
Proposition 30.16. A polynomial \( p \in K\left\lbrack X\right\rbrack \) is irreducible iff \( \left( p\right) \) is a maximal ideal in \( K\left\lbrack X\right\rbrack \) . | Proof. Since \( K\left\lbrack X\right\rbrack \) is an integral domain, for all nonzero polynomials \( p, q \in K\left\lbrack X\right\rbrack ,\deg \left( {pq}\right) = \) \( \deg \left( p\right) + \deg \left( q\right) \), and thus, \( \left( p\right) \neq K\left\lbrack X\right\rbrack \) iff \( \deg \left( p\right) \geq ... | Yes |
Proposition 30.19. Let \( f \in A\left\lbrack X\right\rbrack \) be any polynomial and \( \alpha \in A \) any element of \( A \) . If the result of dividing \( f \) by \( X - \alpha \) is \( f = \left( {X - \alpha }\right) q + r \), then \( r = 0 \) iff \( f\left( \alpha \right) = 0 \), i.e., \( \alpha \) is a root of \... | Proof. We have \( f = \left( {X - \alpha }\right) q + r \), with \( \deg \left( r\right) < 1 = \deg \left( {X - \alpha }\right) \) . Thus, \( r \) is a constant in \( K \), and since \( f\left( \alpha \right) = \left( {\alpha - \alpha }\right) q\left( \alpha \right) + r \), we get \( f\left( \alpha \right) = r \), and ... | Yes |
Proposition 30.20. Let \( f \in A\left\lbrack X\right\rbrack \) be any nonnull polynomial and \( h \geq 0 \) any integer. The following conditions are equivalent.\n\n(1) \( f \) is divisible by \( {\left( X - \alpha \right) }^{h} \) but not by \( {\left( X - \alpha \right) }^{h + 1} \).\n\n(2) There is some \( g \in A\... | Proof. Assume (1). Then, we have \( f = {\left( X - \alpha \right) }^{h}g \) for some \( g \in A\left\lbrack X\right\rbrack \) . If we had \( g\left( \alpha \right) = 0 \) , by Proposition 30.19, \( g \) would be divisible by \( \left( {X - \alpha }\right) \), and then \( f \) would be divisible by \( {\left( X - \alph... | Yes |
Proposition 30.21. Let \( f, g \in A\left\lbrack X\right\rbrack \) be nonnull polynomials, let \( \alpha \in A \), and let \( h \geq 0 \) and \( k \geq 0 \) be the multiplicities of \( \alpha \) with respect to \( f \) and \( g \) . The following properties hold.\n\n(1) If \( l \) is the multiplicity of \( \alpha \) wi... | Proof. (1) We have \( f\left( X\right) = {\left( X - \alpha \right) }^{h}{f}_{1}\left( X\right), g\left( X\right) = {\left( X - \alpha \right) }^{k}{g}_{1}\left( X\right) \), with \( {f}_{1}\left( \alpha \right) \neq 0 \) and \( {g}_{1}\left( \alpha \right) \neq 0 \) . Clearly, \( l \geq \min \left( {h, k}\right) \) . ... | Yes |
Let \( A \) be an integral domain. Let \( f \) be any nonnull polynomial \( f \in A\left\lbrack X\right\rbrack \) and let \( {\alpha }_{1},\ldots ,{\alpha }_{m} \in A \) be \( m \geq 1 \) distinct roots of \( f \) of respective multiplicities \( {k}_{1},\ldots ,{k}_{m} \) . Then, we have\n\n\[ f\left( X\right) = {\left... | Proof. We proceed by induction on \( m \) . The case \( m = 1 \) is obvious in view of Definition 30.12 (which itself, is justified by Proposition 30.20). If \( m \geq 2 \), by the induction hypothesis, we have\n\n\[ f\left( X\right) = {\left( X - {\alpha }_{1}\right) }^{{k}_{1}}\cdots {\left( X - {\alpha }_{m - 1}\rig... | Yes |
Theorem 30.23. Let \( A \) be an integral domain. For every nonnull polynomial \( f \in A\left\lbrack X\right\rbrack \), if the degree of \( f \) is \( n = \deg \left( f\right) \) and \( {k}_{1},\ldots ,{k}_{m} \) are the multiplicities of all the distinct roots of \( f \) (where \( m \geq 0 \) ), then \( {k}_{1} + \cd... | Proof. Immediate from Proposition 30.22. | No |
Let \( A \) be any ring. For every nonnull polynomial \( f \in A\left\lbrack X\right\rbrack ,\alpha \in A \) is a simple root of \( f \) iff \( \alpha \) is a root of \( f \) and \( \alpha \) is not a root of \( {f}^{\prime } \) . | Proof. Since \( \alpha \in A \) is a root of \( f \), we have \( f = \left( {X - \alpha }\right) g \) for some \( g \in A\left\lbrack X\right\rbrack \) . Now, \( \alpha \) is a simple root of \( f \) iff \( g\left( \alpha \right) \neq 0 \) . However, we have \( {f}^{\prime } = g + \left( {X - \alpha }\right) {g}^{\prim... | Yes |
Proposition 30.26. Let \( A \) be any ring. For every nonnull polynomial \( f \in A\left\lbrack X\right\rbrack \), let \( \alpha \in A \) be a root of multiplicity \( k \geq 1 \) of \( f \). Then, \( \alpha \) is a root of multiplicity at least \( k - 1 \) of \( {f}^{\prime } \). If \( A \) is a field of characteristic... | Proof. Since \( \alpha \in A \) is a root of multiplicity \( k \) of \( f \), we have \( f = {\left( X - \alpha \right) }^{k}g \) for some \( g \in A\left\lbrack X\right\rbrack \) and \( g\left( \alpha \right) \neq 0 \). Since\n\n\[ \n{f}^{\prime } = k{\left( X - \alpha \right) }^{k - 1}g + {\left( X - \alpha \right) }... | Yes |
The Hermite interpolation problem has a unique solution of degree \( \leq n \) , where \( n = {n}_{1} + \cdots + {n}_{m + 1} + m \) . | First, we prove that the Hermite interpolation problem has at most one solution. Assume that \( P \) and \( Q \) are two distinct solutions of degree \( \leq n \) . Then, by Proposition 30.26 and the criterion following it, \( P - Q \) has among its roots \( {\alpha }_{1} \) of multiplicity at least \( {n}_{1} + 1,\ldo... | Yes |
Proposition 31.1. Let \( f : E \rightarrow E \) be a linear map on some finite-dimensional vector space \( E \) . Then \( \lambda \in K \) is a zero of the minimal polynomial \( {m}_{f}\left( X\right) \) of \( f \) iff \( \lambda \) is an eigenvalue of \( f \) iff \( \lambda \) is a zero of \( {\chi }_{f}\left( X\right... | Proof. First assume that \( m\left( \lambda \right) = 0 \) (with \( \lambda \in K \), and writing \( m \) instead of \( {m}_{f} \) ). If so, using polynomial division, \( m \) can be factored as \[ m = \left( {X - \lambda }\right) q \] with \( \deg \left( q\right) < \deg \left( m\right) \) . Since \( m \) is the minima... | Yes |
Proposition 31.2. Let \( f : E \rightarrow E \) be a linear map on some finite-dimensional vector space E. If \( f \) diagonalizable, then its minimal polynomial is a product of distinct factors of degree 1. | Proof. If we assume that \( f \) is diagonalizable, then its eigenvalues are all in \( K \), and if \( {\lambda }_{1},\ldots ,{\lambda }_{k} \) are the distinct eigenvalues of \( f \), and then by Proposition 31.1, the minimal polynomial \( m \) of \( f \) must be a product of powers of the polynomials \( \left( {X - {... | Yes |
Proposition 31.3. Let \( W \) be a subspace of \( E \) invariant under the linear map \( f : E \rightarrow E \) (where \( E \) is finite-dimensional). Then the minimal polynomial of the restriction \( f \mid W \) of \( f \) to \( W \) divides the minimal polynomial of \( f \), and the characteristic polynomial of \( f ... | Sketch of proof. The key ingredient is that we can pick a basis \( \left( {{e}_{1},\ldots ,{e}_{n}}\right) \) of \( E \) in which \( \left( {{e}_{1},\ldots ,{e}_{k}}\right) \) is a basis of \( W \) . The matrix of \( f \) over this basis is a block matrix of the form\n\n\[ A = \left( \begin{matrix} B & C \\ 0 & D \end{... | Yes |
Proposition 31.4. If \( W \) is an invariant subspace for \( f \), then for each \( u \in E \), the \( f \) -conductor \( {S}_{f}\left( {u, W}\right) \) is an ideal in \( K\left\lbrack X\right\rbrack \) . | We leave the proof as a simple exercise, using the fact that if \( W \) invariant under \( f \), then \( W \) is invariant under every polynomial \( q\left( f\right) \) in \( {S}_{f}\left( {u, W}\right) \) . | No |
For example, suppose \( f : {\mathbb{R}}^{2} \rightarrow {\mathbb{R}}^{2} \) where \( f\left( {x, y}\right) = \left( {x,0}\right) \) . Observe that \( W = \left\{ {\left( {x,0}\right) \in {\mathbb{R}}^{2}}\right\} \) is invariant under \( f \) . | By representing \( f \) as \( \left( \begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right) \), we see that \( {m}_{f}\left( X\right) = \) \( {\chi }_{f}\left( X\right) = {X}^{2} - X \) . Let \( u = \left( {0, y}\right) \) . Then \( {S}_{f}\left( {u, W}\right) = \left( X\right) \) and we say \( X \) is the conductor of \(... | No |
Proposition 31.5. Let \( f : E \rightarrow E \) be a linear map on a finite-dimensional space \( E \) and assume that the minimal polynomial \( m \) of \( f \) is of the form\n\n\[ m = {\left( X - {\lambda }_{1}\right) }^{{r}_{1}}\cdots {\left( X - {\lambda }_{k}\right) }^{{r}_{k}} \]\n\nwhere the eigenvalues \( {\lamb... | Proof. Observe that (a) and (b) together assert that the conductor of \( u \) into \( W \) is a polynomial of the form \( X - {\lambda }_{i} \) . Pick any vector \( v \in E \) not in \( W \), and let \( g \) be the conductor of \( v \) into \( W \), i.e. \( g\left( f\right) \left( v\right) \in W \) . Since \( g \) divi... | Yes |
Theorem 31.6. Let \( f : E \rightarrow E \) be a linear map on a finite-dimensional space \( E \) . Then \( f \) is diagonalizable iff its minimal polynomial \( m \) is of the form\n\n\[ m = \left( {X - {\lambda }_{1}}\right) \cdots \left( {X - {\lambda }_{k}}\right) \]\n\nwhere \( {\lambda }_{1},\ldots ,{\lambda }_{k}... | Proof. We already showed in Proposition 31.2 that if \( f \) is diagonalizable, then its minimal polynomial is of the above form (where \( {\lambda }_{1},\ldots ,{\lambda }_{k} \) are the distinct eigenvalues of \( f \) ).\n\nFor the converse, let \( W \) be the subspace spanned by all the eigenvectors of \( f \) . If ... | Yes |
Proposition 31.7. Let \( \mathcal{F} \) be a finite commuting family of diagonalizable linear maps on a vector space \( E \) . There exists a basis of \( E \) such that every linear map in \( \mathcal{F} \) is represented in that basis by a diagonal matrix. | Proof. We proceed by induction on \( n = \dim \left( E\right) \) . If \( n = 1 \), there is nothing to prove. If \( n > 1 \), there are two cases. If all linear maps in \( \mathcal{F} \) are of the form \( \lambda \) id for some \( \lambda \in \) \( K \), then the proposition holds trivially. In the second case, let \(... | Yes |
Proposition 31.9. Let \( \mathcal{F} \) be a nonempty finite commuting family of triangulable linear maps on a finite-dimensional vector space \( E \) . There exists a basis of \( E \) such that every linear map in \( \mathcal{F} \) is represented in that basis by an upper triangular matrix. | Proof. Let \( n = \dim \left( E\right) \) . We construct inductively a basis \( \left( {{u}_{1},\ldots ,{u}_{n}}\right) \) of \( E \) such that if \( {W}_{i} \) is the subspace spanned by \( \left( {{u}_{1}\ldots ,{u}_{i}}\right) \), then for every \( f \in \mathcal{F} \) ,\n\n\[ f\left( {u}_{i}\right) = {a}_{1i}^{f}{u... | Yes |
Example 31.2. First let \( f : {\mathbb{R}}^{3} \rightarrow {\mathbb{R}}^{3} \) be defined as \( f\left( {x, y, z}\right) = \left( {y, - x, z}\right) \) . In terms of the standard basis \( f \) is represented by the \( 3 \times 3 \) matrix \( {X}_{f} \mathrel{\text{:=}} \left( \begin{matrix} 0 & - 1 & 0 \\ 1 & 0 & 0 \\... | Using the notation of the preceding proof set\n\n\[ m = {p}_{1}{p}_{2},\;{p}_{1} = {x}^{2} + 1,\;{p}_{2} = x - 1. \]\n\nThen\n\n\[ {g}_{1} = \frac{m}{{p}_{1}} = x - 1,\;{g}_{2} = \frac{m}{{p}_{2}} = {x}^{2} + 1. \]\n\nWe must find \( {h}_{1},{h}_{2} \in \mathbb{R}\left\lbrack x\right\rbrack \) such that \( {g}_{1}{h}_{... | Yes |
For our second example of the primary decomposition theorem let \( f : {\mathbb{R}}^{3} \rightarrow \) \( {\mathbb{R}}^{3} \) be defined as \( f\left( {x, y, z}\right) = \left( {y, - x + z, - y}\right) \), with standard matrix representation \( {X}_{f} = \n\n\( \left( \begin{matrix} 0 & - 1 & 0 \\ 1 & 0 & - 1 \\ 0 & 1 ... | Set\n\n\[ \n{p}_{1} = {x}^{2} + 2,\;{p}_{2} = x,\;{g}_{1} = \frac{{m}_{f}}{{p}_{1}} = x,\;{g}_{2} = \frac{{m}_{f}}{{p}_{2}} = {x}^{2} + 2.\n\]\n\nSince \( \gcd \left( {{g}_{1},{g}_{2}}\right) = 1 \), we use the Euclidean algorithm to find\n\n\[ \n{h}_{1} = - \frac{1}{2}x,\;{h}_{2} = \frac{1}{2},\n\]\n\nsuch that \( {g}... | Yes |
Theorem 31.11. (Primary Decomposition Theorem, Version 2) Let \( f : E \rightarrow E \) be a linear map on the finite-dimensional vector space \( E \) over the field \( K \). If all the eigenvalues \( {\lambda }_{1},\ldots ,{\lambda }_{k} \) of \( f \) belong to \( K \), write\n\n\[ m = {\left( X - {\lambda }_{1}\right... | Proof. Parts (a), (b) and (d) have already been proven in Theorem 31.10, so it remains to prove (c). Since \( {W}_{i} \) is invariant under \( f \), let \( {f}_{i} \) be the restriction of \( f \) to \( {W}_{i} \). The characteristic polynomial \( {\chi }_{{f}_{i}} \) of \( {f}_{i} \) divides \( \chi \left( f\right) \)... | Yes |
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