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Every n-dimensional vector space \( V \) over a field \( \mathbb{k} \) is isomorphic to the coordinate space \( {\mathbb{k}}^{n} \) . The isomorphisms \( F : {\mathbb{k}}^{n} \simeq V \) are in one-to-one correspondence with the bases of \( V \) . | For every isomorphism \( F : {\mathbb{k}}^{n} \rightarrow V \), the images of the standard basis vectors \( {}^{11} \) \( {e}_{i} \in {\mathbb{k}}^{n} \) form a basis in \( V \) . Conversely, for any basis \( {v}_{1},{v}_{2},\ldots ,{v}_{n} \) in \( V \), the map\n\n\[ F : {\mathbb{k}}^{n} \rightarrow V,\;\left( {{x}_{... | Yes |
Example 6.11 (Finite Field Extensions) Let \( \mathbb{k} \subset \mathbb{F} \) be an extension of fields. Then \( \mathbb{F} \) is a vector space over \( \mathbb{k} \). If it has finite dimension \( {\dim }_{\mathbb{k}}\mathbb{F} = d \), then the extension \( \mathbb{k} \subset \mathbb{F} \) is called a finite extensio... | In particular, every finite field \( \mathbb{F} \) of characteristic \( p \) is a finite extension of the prime subfield \( {\mathbb{F}}_{p} \subset \mathbb{F} \). Then \( \left| \mathbb{F}\right| = {p}^{d} \), where \( d = {\dim }_{{\mathbb{F}}_{p}}\mathbb{F} \), by Corollary 6.3. This is a simple conceptual proof of ... | No |
Proposition 6.1 Let \( F : V \mapsto W \) be a linear map and \( K, L \subset V \) two sets of vectors such that \( K \) is a basis in \( \ker F \) and \( K \cup L \) is a basis in \( V \). Then all vectors \( {w}_{e} = F\left( e\right) \), \( e \in L \), are distinct, and they form a basis in \( \operatorname{im}F \).... | Proof Since \( F \) sends an arbitrary vector \( v = \mathop{\sum }\limits_{{f \in K}}{x}_{f}f + \mathop{\sum }\limits_{{e \in L}}{y}_{e}e \in V \) to\n\n\[ \nF\left( v\right) = \mathop{\sum }\limits_{{f \in K}}{x}_{f}F\left( f\right) + \mathop{\sum }\limits_{{e \in L}}{y}_{e}F\left( e\right) = \mathop{\sum }\limits_{{... | Yes |
Corollary 6.4 The following properties of a linear endomorphism \( F : V \rightarrow V \) of a finite-dimensional vector space \( V \) are equivalent:\n\n(1) \( F \) is bijective.\n\n(2) \( \ker F = 0 \) .\n\n(3) \( \operatorname{im}F = V \) . | Proof Properties (2) and (3) are equivalent by the relation (6.18). Since (2) means the injectivity of \( F \), property (1) is equivalent to the simultaneous fulfillment of (2) and (3). | Yes |
Proposition 6.2 For every choice of bases \( \mathbf{u} = \left( {{u}_{1},{u}_{2},\ldots ,{u}_{n}}\right) \) in \( U \) and \( \mathbf{w} = \) \( \left( {{w}_{1},{w}_{2},\ldots ,{w}_{m}}\right) \) in \( W \), the map\n\n\[ \n{\operatorname{Hom}}_{\mathbb{k}}\left( {U, W}\right) \rightarrow {\operatorname{Mat}}_{m \time... | Proof The map (6.21) is linear by Exercise 6.9 and injective, because every linear map \( F : U \rightarrow W \) is completely determined by its action on the basis vectors: as soon the matrix of \( F \) is given, every vector of the form \( v = \sum {u}_{j}{x}_{j} \) will be sent to\n\n\[ \nF\left( v\right) = F\left( ... | No |
Proposition 6.3 For any two finite-dimensional vector subspaces \( U, W \) in an arbitrary vector space \( V \), we have the equality\n\n\[ \n\dim \left( U\right) + \dim \left( W\right) = \dim \left( {U \cap W}\right) + \dim \left( {U + W}\right) .\n\] | Proof Fix some basis \( {e}_{1},{e}_{2},\ldots ,{e}_{k} \) in \( U \cap W \) and extend it to bases in \( U \) and \( W \) by appropriate vectors \( {u}_{1},{u}_{2},\ldots ,{u}_{r} \in U \) and \( {w}_{1},{w}_{2},\ldots ,{w}_{s} \in W \) respectively. It is enough to check that the vectors \( {e}_{1},{e}_{2},\ldots ,{e... | Yes |
Corollary 6.7 For any two subspaces \( U, W \) of a finite-dimensional vector space \( V \) , the inequality\n\n\[ \dim \left( {U \cap W}\right) \geq \dim \left( U\right) + \dim \left( W\right) - \dim \left( V\right) \]\n\nholds. In particular, if \( \dim \left( U\right) + \dim \left( W\right) > \dim V \), then \( U \c... | Proof This follows at once from Proposition 6.3 and the inequality \( \dim \left( {U + W}\right) \leq \) \( \dim V \) . | Yes |
Corollary 6.8 The following conditions on finite-dimensional subspaces \( U, W \subset V \) are equivalent:\n\n(1) \( \dim \left( {U + W}\right) = \dim U + \dim W \) ;\n\n(2) \( U \cap W = 0 \) ;\n\n(3) every vector \( v \in U + W \) has a unique decomposition \( v = u + w \), where \( u \in U \) , \( w \in W \) . | Proof Conditions (1), (2) are equivalent by Proposition 6.3. Let us prove the equivalence of (2) and (3). If there is some nonzero vector \( v \in U \cap W \), then \( 0 + 0 = 0 = v + \left( {-v}\right) \) are two different decompositions of the zero vector as a sum \( u + w, u \in U, w \in W \) . Conversely, the equal... | Yes |
Proposition 6.4 Let the affine subspaces \( \Pi \left( {p, U}\right) ,\Pi \left( {q, U}\right) \) share the same direction subspace \( U \subset V \) . Then the following properties are equivalent:\n\n(1) \( \overrightarrow{pq} \in U \) ,\n\n(2) \( \Pi \left( {p, U}\right) = \Pi \left( {q, U}\right) \) ,\n\n(3) \( \Pi ... | Proof We first check that \( \left( 1\right) \Rightarrow \left( 2\right) \) . If \( \overrightarrow{pq} \in U \), then every point \( q + u, u \in U \), can be written as \( p + w \) for \( w = \overrightarrow{pq} + u \in U \) . Conversely, every point \( p + w, w \in U \), can be written as \( p + u \) for \( u = w - ... | Yes |
Lemma 6.3 Given \( k + 1 \) points \( {p}_{0},{p}_{1},\ldots ,{p}_{k} \) in an affine space \( A \) over a vector space \( V \), the vectors\n\n\[ \n{\overrightarrow{{p}_{0}p}}_{1},{\overrightarrow{{p}_{0}p}}_{2},\ldots ,{\overrightarrow{{p}_{0}p}}_{k} \n\]\n\nare linearly independent in \( V \) if and only if the poin... | Proof The vectors \( {\overrightarrow{p}}_{0}{\overrightarrow{p}}_{1},{\overrightarrow{p}}_{0}{\overrightarrow{p}}_{2},\ldots ,{\overrightarrow{p}}_{0}{\overrightarrow{p}}_{k} \) are linearly related if and only if their linear span has dimension less than \( k \) . The latter is equivalent to the existence of a linear... | Yes |
Proposition 6.5 For any linearly general \( k + 1 \) points in an affine space \( A \), there exists a unique \( k \) -dimensional affine subspace containing these \( k + 1 \) points. If \( \dim A \geq k + 1 \), then the converse statement is also true. | Proof An affine subspace \( {p}_{0} + U \) contains all points \( {p}_{i} \) if and only if all vectors \( \overrightarrow{{p}_{0}{p}_{i}} \) belong to \( U \) . The linear independence of these vectors means that they form a basis in a \( k \) -dimensional vector subspace \( U \subset V \) containing them all. Therefo... | Yes |
Write \( U \subset {\mathbb{k}}^{n} \) for the linear space of solutions of the homogeneous system of linear equations\n\n\[ \n\\begin{cases} {a}_{11}{x}_{1} + {a}_{12}{x}_{2} + \\cdots + {a}_{1n}{x}_{n} & = 0, \\\\ {a}_{21}{x}_{1} + {a}_{22}{x}_{2} + \\cdots + {a}_{2n}{x}_{n} & = 0, \\\\ {a}_{31}{x}_{1} + {a}_{32}{x}_... | For a point \( p \\in {\\mathbb{A}}^{n} = \\mathbb{A}\\left( {\\mathbb{k}}^{n}\\right) \), the affine subspace \( p + U = p + \\ker A \) is the set of solutions of the inhomogeneous system of linear equations \( A\\left( x\\right) = b \) whose right-hand side is given by \( b = A\\left( p\\right) \\in {\\mathbb{k}}^{m}... | Yes |
Lemma 6.4 If a map (6.31) is linear for some \( p \in A \), then it does not depend on the choice of \( p \in A \) . | Proof If \( {D}_{p}\varphi \) is linear, then for every point \( r \in A \) and vector \( u = \overrightarrow{rq} = \overrightarrow{pq} - \overrightarrow{pr} \in U \) ,\n\n\[ \n{D}_{r}\varphi \left( u\right) = \overrightarrow{\varphi \left( r\right) \varphi \left( q\right) } = \overrightarrow{\varphi \left( p\right) \v... | Yes |
Proposition 6.6 Let \( A, B, C \) be affine spaces associated with vector spaces \( U, V \) , W. For two affine maps \( \psi : A \rightarrow B,\varphi : B \rightarrow C \), the composition \( \varphi \circ \psi : A \rightarrow C \) is affine, and \( D\left( {\varphi \circ \psi }\right) = \left( {D\varphi }\right) \circ... | \[ \text{Proof}{D}_{p}\left( {\varphi \circ \psi }\right) : \overrightarrow{pq} \mapsto \overrightarrow{{\varphi \psi }\left( p\right) {\varphi \psi }\left( q\right) } = {D\varphi }\left( \overrightarrow{\psi \left( p\right) \psi \left( q\right) }\right) = {D\varphi } \circ {D\psi }\left( \overrightarrow{pq}\right) \te... | Yes |
An affine endomorphism \( \varphi : \mathbb{A}\left( V\right) \rightarrow \mathbb{A}\left( V\right) \) is bijective if and only if \( {D\varphi } : V \rightarrow V \) is bijective. An affine automorphism \( \varphi \) is a shift if and only if \( {D\varphi } = {\operatorname{Id}}_{V} \) | Both statements follow from the equality \( \varphi \left( {p + v}\right) = \varphi \left( p\right) + {D\varphi }\left( v\right) \) . | No |
Proposition 6.8 Let \( V \) be a vector space, \( U \subset V \) a subspace, and \( R \subset U, S \subset V \smallsetminus U \) two sets of vectors such that \( R \) is a basis in \( U \) and \( R \cup S \) is a basis in \( V \) . Then the congruence classes \( {\left\lbrack w\right\rbrack }_{U}, w \in S \), are disti... | Proof This follows from Proposition 6.1 on p. 135 applied to the quotient map \( V \rightarrow V/U \) . | No |
Example 6.20 (Linear Span as a Quotient Space) The linear span\n\n\\[ \nW = \\operatorname{span}\\left( {{w}_{1},{w}_{2},\\ldots ,{w}_{m}}\\right)\n\\]\n\nof any collection of vectors \\( {w}_{1},{w}_{2},\\ldots ,{w}_{m} \\in V \\) can be viewed as the image of the linear map \\( F : {\\mathbb{k}}^{m} \\rightarrow V \\... | The kernel of this map \\( U = \\ker F \\subset {\\mathbb{k}}^{m} \\) is nothing but the space of linear relations among the vectors \\( {w}_{i} \\) in \\( V \\), because it consists of all rows \\( \\left( {{\\lambda }_{1},{\\lambda }_{2},\\ldots ,{\\lambda }_{m}}\\right) \\in {\\mathbb{k}}^{m} \\) such that \\( {\\la... | Yes |
Lemma 7.1 Let \( E \subset V \) be a basis \( {}^{2} \) of a vector space \( V \) . For a linear form \( \varphi \) : \( V \rightarrow \mathbb{k} \), write \( {\left. \varphi \right| }_{E} : E \rightarrow \mathbb{k} \) for the restriction of \( \varphi \) to \( E \subset V \) . Then the assignment \( \varphi \mapsto {\... | Proof (of Lemma 7.1) We construct the inverse map \( {\mathbb{k}}^{E} \rightarrow {V}^{ * }, f \mapsto \widetilde{f} \) . For a function \( f : E \rightarrow \mathbb{k} \) and vector \( v = \mathop{\sum }\limits_{{e \in E}}{x}_{e}e \), put \( \widetilde{f}\left( v\right) = \mathop{\sum }\limits_{{e \in E}}{x}_{e}f\left... | Yes |
Proposition 7.1 For every basis \( E \subset V \), the set of coordinate forms \( {\left\{ {e}^{ * }\right\} }_{e \in E} \) is linearly independent in \( {V}^{ * } \) . If \( E \) is finite, they form a basis of \( {V}^{ * } \) . In particular, \( \dim V = \dim {V}^{ * } \) in this case. | Proof Evaluation of both sides of the linear relation \( {}^{5}\mathop{\sum }\limits_{{e \in E}}{\lambda }_{e}{e}^{ * } = 0 \) at a basic vector \( w \in E \) leads to the equality \( {\lambda }_{w} = 0 \) . Therefore, the covectors \( {e}^{ * } \) are linearly independent. If \( E \) is finite, then every linear form ... | Yes |
Example 7.3 (Power Series as Linear Forms on Polynomials) The space of polynomials \( \mathbb{k}\left\lbrack x\right\rbrack \) has the standard countable basis formed by monomials \( {x}^{k} \) . By Lemma 7.1, the dual space \( \mathbb{k}{\left\lbrack x\right\rbrack }^{ * } \) is isomorphic to the space of sequences \(... | \[ f = {f}_{0} + {f}_{1}x + {f}_{2}{x}^{2} + \cdots \in \mathbb{k}\llbracket x\rrbracket \] maps \( \widetilde{f} : {a}_{0} + {a}_{1}x + \cdots + {a}_{m}{x}^{m} \mapsto {f}_{0}{a}_{0} + {f}_{1}{a}_{1} + \cdots + {f}_{m}{a}_{m} \) . For example, for every \( \alpha \in \mathbb{k} \), the evaluation form \( {\operatornam... | Yes |
Theorem 7.1 The canonical map (7.3) is injective. If \( \dim V < \infty \), then it is an isomorphism. The latter means that every linear form \( \xi : {V}^{ * } \rightarrow \mathbb{k} \) coincides with evaluation \( {\mathrm{{ev}}}_{v} : {V}^{ * } \rightarrow \mathbb{k} \) for some vector \( v \in V \) uniquely determ... | Proof Injectivity means that for every nonzero vector \( v \in V \), there exists some covector \( \varphi \in {V}^{ * } \) such that \( {\operatorname{ev}}_{v}\left( \varphi \right) = \varphi \left( v\right) \neq 0 \) . By Theorem 6.1 on p. 132, the vector \( v \) can be included in some basis \( E \) of \( V \) . The... | Yes |
Every \( n + 1 \) distinct constants \( {a}_{0},{a}_{1},\ldots ,{a}_{n} \in \) \( \mathbb{k} \) produce \( n + 1 \) evaluation forms on the space \( \mathbb{k}{\left\lbrack x\right\rbrack }_{ \leq n} \) of polynomials of degree at most \( n \) : | The polynomial \( {f}_{i}\left( x\right) = \mathop{\prod }\limits_{{v \neq i}}\left( {x - {a}_{v}}\right) \) vanishes at all points \( {a}_{v} \) except for \( {a}_{i} \), where \( {f}_{i}\left( {a}_{i}\right) \neq 0 \) . Therefore, the polynomials \( {v}_{i}\left( x\right) = {f}_{i}\left( x\right) /{f}_{i}\left( {a}_{... | Yes |
Example 7.5 (Taylor’s Formula) Assume that char \( \mathbb{k} = 0 \), fix some point \( a \in \mathbb{k} \) , and for \( 0 \leq k \leq n \) write\n\n\[ \n{\varphi }_{k} = {\operatorname{ev}}_{a} \circ \frac{{d}^{k}}{d{x}^{k}} : \mathbb{k}{\left\lbrack x\right\rbrack }_{ \leq n} \rightarrow \mathbb{k},\;f \mapsto {f}^{\... | \[ \ng\left( x\right) = g\left( a\right) + {g}^{\prime }\left( a\right) \cdot \left( {x - a}\right) + {g}^{\prime \prime }\left( a\right) \cdot \frac{{\left( x - a\right) }^{2}}{2} + \cdots + {g}^{\left( n\right) }\left( a\right) \cdot \frac{{\left( x - a\right) }^{n}}{n!}.\n\]\n\n(7.5)\n\nThe expansion (7.5) is called... | Yes |
Proposition 7.2 (Perfect Pairing) The following properties of a pairing \( \langle * , * \rangle \) : \( U \times W \rightarrow \mathbb{k} \) between finite-dimensional vector spaces \( U, W \) are equivalent:\n\n(1) For every nonzero \( u \in U \), there is some \( w \in W \), and for every nonzero \( w \in W \) , the... | Proof Since \( \langle v, w\rangle \) is bilinear, both maps from (2),(3) are well defined and linear. Condition (1) says that the both are injective. Therefore, if (1) holds, then \( \dim U \leq \) \( \dim {W}^{ * } \) and \( \dim W \leq \dim {U}^{ * } \) . Since \( \dim U = \dim {U}^{ * } \) and \( \dim W = \dim {W}^... | No |
Example 7.6 (Contraction Between Vectors and Covectors) Evaluation of a covector \( \varphi \in {V}^{ * } \) at the vector \( v \in V \) gives a perfect pairing | \[ \langle v,\varphi \rangle \triangleq \varphi \left( v\right) = {\operatorname{ev}}_{v}\left( \varphi \right) \] between dual finite-dimensional vector spaces \( V \) and \( {V}^{ * } \). It is called a contraction of a vector with a covector. The notation \( \langle v,\varphi \rangle \) emphasizes the symmetry betwe... | Yes |
Proposition 7.3 \( \dim U + \dim \operatorname{Ann}U = \dim V \) for every subspace \( U \subset V \) . | Proof Let the vectors \( {u}_{1},{u}_{2},\ldots ,{u}_{k} \) be a basis of \( U \) and suppose that the vectors \( {w}_{1},{w}_{2},\ldots ,{w}_{m} \) complete them to a basis in \( V \) . Therefore, \( \dim V = k + m \) . Write\n\n\[ \n{u}_{1}^{ * },{u}_{2}^{ * },\ldots ,{u}_{k}^{ * },{w}_{1}^{ * },{w}_{2}^{ * },\ldots ... | Yes |
Corollary 7.1 Ann Ann \( \left( U\right) = U \) for every subspace \( U \subset V \) . | Proof By definition, \( U \subset \operatorname{Ann}\operatorname{Ann}\left( U\right) \) . At the same time, Proposition 7.3 implies that \( \dim \mathrm{{Ann}}\operatorname{Ann}U = \dim {V}^{ * } - \dim \mathrm{{Ann}}U = \dim {V}^{ * } - \dim V + \dim U = \dim U \) . | Yes |
Corollary 7.2 \( \dim U + \dim \operatorname{Ann}U = \dim V \) and \( \operatorname{Ann}\operatorname{Ann}\left( U\right) = U \) for every subspace \( U \subset {V}^{ * } \) as well. | Proof Apply Proposition 7.3 and Corollary 7.1 to the dual space \( {V}^{ * } \) instead of \( V \) and use the canonical identification \( {V}^{* * } \simeq V \) . | No |
Theorem 7.2 The correspondence \( U \leftrightarrow \operatorname{Ann}\left( U\right) \) is a bijection between the subspaces of complementary \( {}^{10} \) dimensions in \( V \) and in \( {V}^{ * } \) . This bijection reverses the inclusions\n\n\[ U \subset W \Leftrightarrow \operatorname{Ann}U \supset \operatorname{A... | Proof Write \( \mathcal{S}\left( V\right) \) for the set of all vector subspaces in \( V \) . The equality Ann \( \operatorname{Ann}\left( U\right) = U \) means that the maps sending each subspace to its annihilator\n\n\[ S\left( V\right) \overset{U \mapsto \operatorname{Ann}U}{\overbrace{\xrightarrow[{\text{ Ann }W \l... | Yes |
Proposition 7.4 Let \( V \) be an arbitrary \( {}^{11} \) vector space \( V \). For every subspace \( U \subset \) \( V \), the restriction map\n\n\[ \n{r}_{U} : {V}^{ * } \rightarrow {U}^{ * },{\left. \;\varphi \mapsto \varphi \right| }_{U}\n\]\n\n(7.8)\n\n\( {}^{10} \) That is, between \( k \) -dimensional and \( \le... | Proof Fix two disjoint sets of vectors \( E \subset U \) and \( F \subset V \) such that \( E \) is a basis of \( U \) and \( E \sqcup F \) is a basis of \( V \). For every linear form \( \xi : U \rightarrow \mathbb{k} \), write \( \widetilde{\xi } : V \rightarrow \mathbb{k} \) for the linear form that sends each basic... | Yes |
Proposition 7.5 \( \ker {F}^{ * } = \operatorname{Ann}\operatorname{im}F \) and \( \operatorname{im}{F}^{ * } = \operatorname{Ann}\ker F \) . | The first equality follows at once from the definition of \( {F}^{ * } \) written in the form (7.12):\n\n\[ \xi \in \operatorname{Ann}\operatorname{im}F \Leftrightarrow \forall v \in V\langle {Fv},\xi \rangle = 0 \Leftrightarrow \forall v \in V\left\langle {v,{F}^{ * }\xi }\right\rangle = 0 \]\n\n\[ \Leftrightarrow {F}... | Yes |
Theorem 7.3 For every matrix \( A \in {\operatorname{Mat}}_{m \times n}\left( \mathbb{k}\right) \), the linear span of its rows in \( {\mathbb{k}}^{n} \) and the linear span of its columns in \( {\mathbb{k}}^{m} \) have equal dimensions. | Proof Write \( F : {\mathbb{k}}^{n} \rightarrow {\mathbb{k}}^{m} \) for the linear map having the matrix \( A \) in the standard bases of \( {\mathbb{k}}^{n} \) and \( {\mathbb{k}}^{m} \) . Then the linear span of the columns of \( A \) is the image of \( F \) , whereas the linear span of rows is the image of the dual ... | Yes |
Theorem 7.4 (Capelli-Fontené-Frobenius-Kronecker-Rouché Theorem) \( {}^{12} \) The system of linear equations\n\n\[ \left\{ \begin{matrix} {a}_{11}{x}_{1} + {a}_{12}{x}_{2} + \cdots + {a}_{1n}{x}_{n} = {b}_{1}, \\ {a}_{21}{x}_{1} + {a}_{22}{x}_{2} + \cdots + {a}_{2n}{x}_{n} = {b}_{2}, \\ {a}_{31}{x}_{1} + {a}_{32}{x}_{... | Proof The solvability of the system \( A\left( x\right) = b \) means that the right-hand-side column \( b \) lies in the linear span of the columns of the left-hand-side matrix \( A = \) \( \left( {a}_{ij}\right) \) . This happens if and only if the attachment of the column \( b \) to the matrix \( A \) does not change... | Yes |
Corollary 7.4 The solutions of a system of homogeneous linear equations\n\n\\[ \n\\begin{cases} {a}_{11}{x}_{1} + {a}_{12}{x}_{2} + \\cdots + {a}_{1n}{x}_{n} & = 0, \\\\ {a}_{21}{x}_{1} + {a}_{22}{x}_{2} + \\cdots + {a}_{2n}{x}_{n} & = 0, \\\\ {a}_{31}{x}_{1} + {a}_{32}{x}_{2} + \\cdots + {a}_{3n}{x}_{n} & = 0, \\\\ & ... | Proof Write \\( F : {\\mathbb{k}}^{n} \\rightarrow {\\mathbb{k}}^{m} \\) for the linear map with matrix \\( A \\) . Then the solution subspace is nothing but \\( \\ker F \\), and \\( \\dim \\ker F = n - \\dim \\operatorname{im}F = n - \\operatorname{rk}A \\) . | Yes |
Example 8.1 (Endomorphism Algebra of a Vector Space) The composition of two linear maps \( G : U \rightarrow V, F : V \rightarrow W \) is linear, because | \[ {FG}\left( {{\lambda u} + {\mu w}}\right) = F\left( {{\lambda G}\left( u\right) + {\mu G}\left( w\right) }\right) = {\lambda FG}\left( u\right) + {\mu FG}\left( w\right) . \] Considered as a binary operation, the composition map \[ \operatorname{Hom}\left( {V, W}\right) \times \operatorname{Hom}\left( {U, V}\right) ... | Yes |
Example 8.3 (Algebraicity of Linear Endomorphisms) Since \( {\dim }_{\mathbb{k}}\operatorname{End}\left( V\right) = {n}^{2} \) , the iterations \( {F}^{0},{F}^{1},{F}^{2},\ldots ,{F}^{{n}^{2}} \) of a linear endomorphism \( F \in \operatorname{End}\left( V\right) \) are linearly related. Therefore, \( F \) satisfies so... | In Sect. 9.6.3 on p. 222 we will show \( {}^{6} \) that \( F \) can actually be annihilated by an appropriate polynomial of degree \( n \) . | No |
Example 8.4 (Annihilating Polynomial of a \( 2 \times 2 \) Matrix) Let us show that every \( 2 \times 2 \) matrix satisfies a quadratic equation: | \[ F = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \;\text{ and }\;{F}^{2} = \left( \begin{array}{ll} {a}^{2} + {bc} & {ab} + {bd} \\ {ca} + {dc} & {cb} + {d}^{2} \end{array}\right) = \left( \begin{array}{ll} {a}^{2} + {bc} & b\left( {a + d}\right) \\ c\left( {a + d}\right) & {cb} + {d}^{2} \end{array}\r... | Yes |
Example 8.5 (Invertible \( 2 \times 2 \) Matrices) It follows from (8.5) that for every \( F \in \) \( {\operatorname{Mat}}_{2}\left( \mathbb{k}\right) \), the equality\n\n\[ \n\det \left( F\right) \cdot E = \operatorname{tr}\left( F\right) \cdot F - {F}^{2} = F \cdot \left( {\operatorname{tr}\left( F\right) E - F}\rig... | \[ \n\det \left( F\right) \cdot {F}^{-1} = \operatorname{tr}\left( F\right) \cdot E - F = \left( \begin{matrix} d & - b \\ - c & a \end{matrix}\right) .\n\]\n\n(8.6)\n\nThis forces \( \det F \) to be nonzero, because otherwise, we would get the zero matrix on the left-hand side and therefore \( a = b = c = d = 0 \) on ... | Yes |
Lemma 8.1 Let a collection of vectors \( \mathbf{v} = \left( {{v}_{1},{v}_{2},\ldots ,{v}_{n}}\right) \) be a basis of \( V \) . Then the collection \( u = v{C}_{vu} \) is a basis of \( V \) if and only if the transition matrix \( {C}_{vu} \) is invertible. In this case, \( {C}_{vu}^{-1} = {C}_{uv} \) . | Proof If \( \mathbf{u} \) is a basis, then the vectors \( \mathbf{e} \) are linearly expressed through \( \mathbf{u} \) . It follows from (8.11) that \( {C}_{ee} = {C}_{eu}{C}_{ue} \) and \( {C}_{uu} = {C}_{ue}{C}_{eu} \) . Since the transition matrix from a collection of vectors to a basis is uniquely determined by th... | Yes |
If the vectors \( \mathbf{w} = \) \( \left( {{w}_{1},{w}_{2},\ldots ,{w}_{m}}\right) \) are expressed through the basis \( \mathbf{e} = \left( {{e}_{1},{e}_{2},\ldots ,{e}_{n}}\right) \) as \( \mathbf{w} = \mathbf{e}{C}_{\mathbf{e}\mathbf{w}} \) and the vectors \( v = e{C}_{ev} \) form another basis, then the transitio... | \[ {C}_{vw} = {C}_{ve}{C}_{ew} = {C}_{ev}^{-1}{C}_{vw} \] | Yes |
Exercise 8.7 Verify this equality. | For every matrix \( M \in {\operatorname{Mat}}_{r \times s}\left( \mathbb{k}\right) \), the equality \( F\left( {vM}\right) = F\left( v\right) M \) holds. | No |
Lemma 8.2 Each matrix \( A \in {\operatorname{Mat}}_{m \times n}\left( \mathbb{k}\right) \) can be transformed to some reduced echelon matrix by means of a finite sequence of elementary row operations. | Proof We split the reduction procedure into \( n \) steps, where \( n \) is a number of columns. Assume inductively that after the \( \left( {k - 1}\right) \) th step, a submatrix formed by the left \( k - 1 \) columns is in reduced echelon form \( {}^{15} \) and has \( s \) nonzero rows. Note that \( 0 \leq s \leq k -... | Yes |
Example 8.8 (How It Works) Let us transform the matrix\n\n\\[ A = \\left( \\begin{array}{rrrrr} 2 & - 4 & - 8 & 2 & - 4 \\ - 1 & 1 & 3 & 0 & 1 \\ - 1 & - 1 & 1 & 2 & - 1 \\ - 1 & 0 & 2 & 1 & 1 \\end{array}\\right) \\in {\\operatorname{Mat}}_{4 \\times 5}\\left( \\mathbb{Q}\\right) \\] | \n\n(8.16)\n\n\\( {}^{15} \\) For \\( k = 1 \\) this means nothing. to reduced echelon form. Multiply the bottom row by -1 , and then swap it with the first row:\n\n\\[\n\\left( \\begin{array}{rrrrr} 1 & 0 & - 2 & - 1 & - 1 \\ - 1 & 1 & 3 & 0 & 1 \\ - 1 & - 1 & 1 & 2 & - 1 \\ 2 & - 4 & - 8 & 2 & - 4 \\end{array}\\right... | Yes |
Since the elementary row operations do not change the linear span of the rows, the nonzero rows of the resulting reduced echelon matrix form a basis in the linear span of rows of the original matrix. | For example, the computation from Example 8.8 shows that a subspace \( U \subset {\mathbb{Q}}^{5} \) spanned by the rows of the matrix \[ \left( \begin{array}{rrrrr} 2 & - 4 & - 8 & 2 & - 4 \\ - 1 & 1 & 3 & 0 & 1 \\ - 1 & - 1 & 1 & 2 & - 1 \\ - 1 & 0 & 2 & 1 & 1 \end{array}\right) \] has dimension 3 , and the rows of t... | Yes |
For every \( r \) -dimensional subspace \( U \subset {\mathbb{k}}^{n} \), the standard basis of \( {\mathbb{k}}^{n} \) can be decomposed into two disjoint sets \( \left\{ {{e}_{{i}_{1}},{e}_{{i}_{2}},\ldots ,{e}_{{i}_{n - r}}}\right\} \sqcup \left\{ {{e}_{{j}_{1}},{e}_{{j}_{2}},\ldots ,{e}_{{j}_{r}}}\right\} = \) \( \l... | Proof For every basis \( {w}_{1},{w}_{2},\ldots ,{w}_{r} \) of \( U \), the exchange lemma \( {}^{17} \) allows us to replace some \( r \) vectors \( {e}_{{j}_{1}},{e}_{{j}_{2}},\ldots ,{e}_{{j}_{r}} \) of the standard basis in \( {\mathbb{k}}^{n} \) by vectors \( {w}_{v} \) in such a way that\n\n\[ \n{w}_{1},{w}_{2},\... | Yes |
Let \( R \) be a reduced echelon matrix of shape \( {}^{19}J \) . Write \( U \) for the linear span of its rows. Then the rows of \( R \) surely satisfy condition (4) of Proposition 8.1. Therefore, \( U \) is isomorphically projected onto the coordinate subspace \( {E}_{J} \) along the complementary coordinate subspace... | For example, one more consequence of the computation made in (8.19) is that the classes of vectors \( {e}_{3} = \left( {0,0,1,0,0}\right) \) and \( {e}_{4} = \left( {0,0,0,1,0}\right) \) form a basis in the quotient space \( {\mathbb{Q}}^{5}/U \), where \( U \subset {\mathbb{Q}}^{5} \) is the subspace spanned by the ro... | Yes |
Example 8.12 (Solution of an Arbitrary System of Linear Equations) Given an arbitrary system of linear equations\n\n\\[ \n\\left\\{ \\begin{matrix} {a}_{11}{x}_{1} + {a}_{12}{x}_{2} + \\cdots + {a}_{1n}{x}_{n} = {b}_{1}, \\\\ {a}_{21}{x}_{1} + {a}_{22}{x}_{2} + \\cdots + {a}_{2n}{x}_{n} = {b}_{2}, \\\\ {a}_{31}{x}_{1} ... | Thus, Gauss's method reduces a system of the form (8.25) to an equivalent system with reduced echelon augmented matrix \\( \\widetilde{R} = \\left\\lceil {R \\mid \\beta }\\right\\rceil \\in {\\operatorname{Mat}}_{r \\times \\left( {n + 1}\\right) }\\left( \\mathbb{k}\\right) \\), where \\( r = \\) \\( \\operatorname{r... | Yes |
Example 8.13 (Graphs of Linear Maps) Let a linear subspace \( U \subset {\mathbb{k}}^{n} \) and the decomposition \( {\mathbb{k}}^{n} = {E}_{I} \oplus {E}_{J} \) satisfy the conditions of Proposition 8.1. Then \( U \) can be viewed as the graph of the linear map \( {f}_{U} = {p}_{I} \circ {p}_{J}^{-1} : {E}_{J} \righta... | Let the rows \( {u}_{1},{u}_{2},\ldots ,{u}_{r} \) of the reduced echelon matrix \( R \) form a basis of \( U \) . Then \( {p}_{J}^{-1}\left( {e}_{{j}_{v}}\right) = {u}_{v} \), and therefore \( {f}_{U}\left( {e}_{{j}_{v}}\right) \) is the \( v \) th row of the submatrix \( {R}_{I} \subset R \) formed by the columns \( ... | Yes |
Example 8.14 Let us analyze whether the matrix\n\n\[ A = \left( \begin{array}{rrrr} 6 & 3 & - 2 & 1 \\ 1 & 4 & 1 & 1 \\ 1 & 1 & 3 & - 1 \\ - 1 & 0 & - 2 & 1 \end{array}\right) \]\n\nis invertible, and if it is, compute the inverse by Gaussian row elimination in the extended matrix\n\n\[ \left( \begin{array}{rrrrrrrr} 6... | Change the sign of the bottom row and swap the top row with the bottom:\n\n\[ \left( \begin{array}{rrrrrrrr} 1 & 0 & 2 & - 1 & 0 & 0 & 0 & - 1 \\ 1 & 4 & 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 3 & - 1 & 0 & 0 & 1 & 0 \\ 6 & 3 & - 2 & 1 & 1 & 0 & 0 & 0 \end{array}\right) . \]\n\nAnnihilate the first column below the first row... | Yes |
Example 8.15 (Solution of a System of Linear Equations Revisited) Using matrix notation, we can write a system of \( n \) linear equations in \( n \) unknowns,\n\n\[ \n\\begin{cases} {a}_{11}{x}_{1} + {a}_{12}{x}_{2} + \\cdots + {a}_{1n}{x}_{n} & = {b}_{1}, \\\\ {a}_{21}{x}_{1} + {a}_{22}{x}_{2} + \\cdots + {a}_{2n}{x}... | It can be found via Gaussian elimination: if we transform the augmented matrix \( A\\left| b\\right| \) to reduced echelon form \( E\\left| c\\right| \), then we get \( c = {A}^{-1}b \) . If we need to solve several systems of equations having the same coefficient matrix \( A \) and different right-hand sides \( b \), ... | Yes |
Over an arbitrary ring \( R \), an upper triangular matrix\n\n\[ \left( \begin{array}{ll} a & b \\ 0 & d \end{array}\right) \]\n\nis invertible if and only if both diagonal elements \( a, d \) are invertible in \( R \) . | Indeed, the equality\n\n\[ \left( \begin{array}{ll} a & b \\ 0 & d \end{array}\right) \left( \begin{array}{ll} x & y \\ z & w \end{array}\right) = \left( \begin{matrix} {ax} + {bz} & {ay} + {bw} \\ {dz} & {dw} \end{matrix}\right) = \left( \begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right) \]\n\nleads to \( {dw} = 1 \)... | Yes |
Lemma 9.1 Every volume form vanishes on linearly related \( {v}_{1},{v}_{2},\ldots ,{v}_{n} \) ; every volume form is linear in each argument,\n\n\[ \omega \left( {\ldots ,{\lambda v} + {\mu w},\ldots }\right) = {\lambda \omega }\left( {\ldots, v,\ldots }\right) + {\mu \omega }\left( {\ldots, w,\ldots }\right) ; \]\n\n... | Proof If vectors \( {v}_{1},{v}_{2},\ldots ,{v}_{n} \) are linearly related, then one of them is a linear combination of the others, say \( {v}_{1} = {\lambda }_{2}{v}_{2} + \cdots + {\lambda }_{n}{v}_{n} \) . Then by the first property of a volume form,\n\n\[ \omega \left( {{v}_{1},{v}_{2},\ldots ,{v}_{n}}\right) = \o... | Yes |
A volume form on an \( n \) -dimensional vector space \( V \) is skew-symmetric \( n \) -linear by Lemma 9.1. Conversely, every skew-symmetric multilinear form of \( n \) arguments satisfies both properties from Definition 9.1 on p. 206 and therefore produces a volume form on \( V \) . | Indeed, the second property is just a part of linearity, and the first property follows from linearity and skew-symmetry:\n\n\[ \omega \left( {\ldots ,{v}_{i} + \lambda {v}_{j},\ldots ,{v}_{j},\ldots }\right) \]\n\n\[ = \omega \left( {\ldots ,{v}_{i},\ldots ,{v}_{j},\ldots }\right) + {\lambda \omega }\left( {\ldots ,{v... | Yes |
Lemma 9.2 For each transposition \( {s}_{ij} \) and arbitrary \( g \in {S}_{n} \), the inversion numbers \( I\left( g\right) \) and \( I\left( {g{s}_{ij}}\right) \) have opposite parities. | Proof Let \( i < j \) . Then the permutations\n\n\[ g = \left( {{g}_{1},\ldots ,{g}_{i - 1},{\mathbf{g}}_{i},{g}_{i + 1},\ldots ,{g}_{i - 1},{\mathbf{g}}_{j},{g}_{j + 1},\ldots ,{g}_{n}}\right) ,\]\n\n(9.4)\n\n\[ g{s}_{ij} = \left( {{g}_{1},\ldots ,{g}_{i - 1},{\mathbf{g}}_{j},{g}_{i + 1},\ldots ,{g}_{i - 1},{\mathbf{g... | Yes |
Corollary 9.1 (Sign of a Permutation) There exists a unique map\n\n\[ \n\operatorname{sgn} : {S}_{n} \rightarrow \{ + 1, - 1\} \n\]\n\nsuch that \( \operatorname{sgn}\left( \mathrm{{Id}}\right) = 1,\operatorname{sgn}\left( {s}_{ij}\right) = - 1 \) for every transposition \( {s}_{ij} \), and\n\n\[ \n\operatorname{sgn}\l... | Proof If \( g \in {S}_{n} \) is a composition of \( k \) transpositions, then \( {\left( -1\right) }^{I\left( g\right) } = {\left( -1\right) }^{k} \) by Lemma 9.2. | No |
Theorem 9.1 For a commutative ring \( K \) with unit there exists a unique, up to a constant factor, nonzero skew-symmetric \( n \) -linear form \( \omega \) on the coordinate module \( {K}^{n} \) . Its value on an arbitrary collection of vectors \( \mathbf{v} = \left( {{v}_{1},{v}_{2},\ldots ,{v}_{n}}\right) \), which... | Proof (of Theorem 9.1) Let us substitute \( {v}_{j} = \mathop{\sum }\limits_{{i = 1}}^{n}{e}_{i} \cdot {c}_{ij} \) in \( \omega \left( {{v}_{1},{v}_{2},\ldots ,{v}_{m}}\right) \) . By the multilinearity of \( \omega \), we get\n\n\[ \omega \left( {{v}_{1},{v}_{2},\ldots ,{v}_{n}}\right) = \omega \left( {\mathop{\sum }\... | No |
Proposition 9.1 Write \( {v}_{1},{v}_{2},\ldots ,{v}_{n} \) for the columns of a matrix \( C \in {\operatorname{Mat}}_{n}\left( K\right) \) and consider them as vectors in \( {K}^{n} \) . Then the function \( \det \left( {{v}_{1},{v}_{2},\ldots ,{v}_{n}}\right) \overset{\text{ def }}{ = }\det C \) is linear in each arg... | Proof Each of the \( n \) ! products in the expansion of det \( C \) contains exactly one factor taken from the \( j \) th column of \( C \) . Hence, \( \det C \) is linear in \( {v}_{j} \) . To prove skew-symmetry, let \( {v}_{i} = {v}_{j} \) and collect pairs of products corresponding to the pairs of permutations \( ... | Yes |
Proposition 9.2 \( \\det \\left( {AB}\\right) = \\det \\left( A\\right) \\cdot \\det \\left( B\\right) \) for every \( A, B \\in {\\operatorname{Mat}}_{n}\\left( K\\right) \), where \( K \) is any commutative ring. | Proof Consider the difference \( \\det \\left( {AB}\\right) - \\det \\left( A\\right) \\cdot \\det \\left( B\\right) \) as a polynomial with integer coefficients in \( 2{n}^{2} \) variables \( {a}_{ij} \) and \( {b}_{ij} \) . It is sufficient to check that it is the zero polynomial.\n\nTherefore, to verify the identity... | Yes |
Given two square matrices \( A, B \in \) \( {\operatorname{Mat}}_{n}\left( K\right) \), consider two commuting scalar variables \( x, y \) and form the matrix \( x \cdot A + y \cdot B \) with elements \( x{a}_{ij} + y{b}_{ij} \in K\left\lbrack {x, y}\right\rbrack \) . Its determinant \( \det \left( {x \cdot A + y \cdot... | \n\[
\mathop{\sum }\limits_{{IJ}}{\left( -1\right) }^{\left| I\right| + \left| J\right| }{a}_{IJ}{b}_{\bar{I}\bar{J}} = \operatorname{tr}\left( {{\mathcal{A}}_{m}{\mathcal{B}}_{m}^{ \vee }}\right) ,
\]
\n(9.24)
\nwhere the left summation runs through all increasing multi-indices \( I, J \subset \) \( \{ 1,2,\ldots, n\}... | Yes |
Example 9.4 (Principal Minors and Trace) For \( x = 1, y = t, B = E \) , formula (9.24) gives the following expansion: | \[ \det \left( {{tE} + A}\right) = {t}^{n} + \mathop{\sum }\limits_{{m = 1}}^{n}{t}^{n - m} \cdot \mathop{\sum }\limits_{{\# I = m}}{a}_{II} \] \[ = {t}^{n} + {t}^{n - 1} \cdot \mathop{\sum }\limits_{i}{a}_{ii} + {t}^{n - 1} \cdot \mathop{\sum }\limits_{{i < j}}\left( {{a}_{ii}{a}_{jj} - {a}_{ij}{a}_{ji}}\right) + \cdo... | Yes |
Proposition 9.3 Over a commutative ring \( K \) with unit, a matrix \( A \in {\operatorname{Mat}}_{n}\left( K\right) \) is invertible if and only if \( \det A \) is invertible in \( K \) . In this case, \( {A}^{-1} = {A}^{ \vee }/\det A \) . | Proof If \( A \) is invertible, then the matrix equality \( A{A}^{-1} = E \) forces \( \det \left( A\right) \det \left( {A}^{-1}\right) = \) 1 in \( K \) . Therefore, \( \det A \) is invertible in \( K \) . Conversely, if \( \det A \) is invertible, then formula (9.29) says that \( {A}^{-1} = {A}^{ \vee }/\det A \) . | Yes |
Example 9.5 For \( 2 \times 2 \) and \( 3 \times 3 \) matrices of determinant 1, we get\n\n\[{\left( \begin{array}{ll} a & b \\ c & d \end{array}\right) }^{-1} = \left( \begin{matrix} d & - b \\ - c & a \end{matrix}\right)\]\n\n\[{\left( \begin{array}{lll} {a}_{11} & {a}_{12} & {a}_{13} \\ {a}_{21} & {a}_{22} & {a}_{23... | For matrices of arbitrary invertible determinant, all matrix elements on the righthand sides must be divided by this determinant. | Yes |
Theorem 9.2 (Cayley-Hamilton Identity) \( {\chi }_{A}\left( A\right) = 0 \) in \( {\operatorname{Mat}}_{n}\left( K\right) \) . | Proof Formula (9.29) written in the matrix algebra \( {\operatorname{Mat}}_{n}\left( {K\left\lbrack t\right\rbrack }\right) \) for the matrix \( {tE} - A \) says that\n\n\[ \det \left( {{tE} - A}\right) \cdot E = \left( {{tE} - A}\right) \cdot {\left( tE - A\right) }^{ \vee }.\]\n\n(9.30)\n\nEach \( B \in {\operatornam... | Yes |
Example 9.6 Each \( 2 \times 2 \) matrix \( A \) satisfies the quadratic equation | \[ {A}^{2} - \operatorname{tr}\left( A\right) \cdot A + \det \left( A\right) \cdot E = 0. \] | Yes |
Proposition 9.4 (Cramer’s Rule I) The vectors \( {v}_{1},{v}_{2},\ldots ,{v}_{n} \in {K}^{n} \) form a basis of \( {K}^{n} \) if and only if \( \det \left( {{v}_{1},{v}_{2},\ldots ,{v}_{n}}\right) \) is invertible in \( K \) . In this case, for every vector \( w = {x}_{1}{v}_{1} + {x}_{2}{v}_{2} + \cdots + {x}_{n}{v}_{... | Proof We know from Lemma 8.1 on p. 181 that the vectors \( {v}_{1},{v}_{2},\ldots ,{v}_{n} \) form a basis of \( {K}^{n} \) if and only if the transition matrix \( C \) from those vectors to the standard basis is invertible.\n\nBy Proposition 9.3, the invertibility of \( C \) is equivalent to the invertibility of \( \d... | Yes |
Corollary 9.6 For every \( w \in {K}^{n} \) and invertible \( A \in {\operatorname{Mat}}_{n}\left( K\right) \), the linear equation \( {Ax} = w \) in the unknown vector \( x \in {K}^{n} \) has a unique solution. The coordinates of this solution are given by Cramer’s rule (9.31), where \( {v}_{1},{v}_{2},\ldots ,{v}_{n}... | Proof The system \( {Ax} = w \) describes the coefficients of the linear expression \( w = \) \( \sum {v}_{i}{x}_{i} \) | No |
Proposition 9.5 (Cramer's Rule II) Given a system of \( n \) linear homogeneous equations in \( n + 1 \) unknowns \( \left( {{x}_{0},{x}_{1},\ldots ,{x}_{n}}\right) \) ,\n\n\[ \n\begin{cases} {a}_{10}{x}_{0} + {a}_{11}{x}_{1} + \cdots + {a}_{1n}{x}_{n} & = 0, \\ {a}_{20}{x}_{0} + {a}_{21}{x}_{1} + \cdots + {a}_{2n}{x}_... | Proof Let us attach a second copy of the \( i \) th row to the top of the matrix \( A \) . We get an \( \left( {n + 1}\right) \times \left( {n + 1}\right) \) matrix with zero determinant. The cofactor expansion of this determinant through the top row leads to the equality\n\n\[ \n{a}_{i0}{A}_{0} - {a}_{i1}{A}_{1} + \cd... | Yes |
Corollary 9.7 Let \( K = \mathbb{k} \) be a field. Then equations (9.32) are linearly independent if and only if Cramer's rule II produces a nonzero solution. In this case, all solutions of the system (9.32) are proportional to the Cramer’s-rule solution. | Proof If the rows of \( A \) are linearly related, then all \( {A}_{i} \) are equal to 0 . If \( \mathrm{{rk}}A = n \) , then by Exercise 8.9 on p. 187, there should be some \( n \times n \) submatrix of rank \( n \) in \( A \) . Then its determinant \( {A}_{i} \) is nonzero. This proves the first statement. If it hold... | No |
Problem 9.13 (Kirchhoff’s Matrix Tree Theorem) For a connected graph \( \Gamma \) with \( n \) vertices numbered \( 1,2,\ldots, n \), consider the square matrix \( A = A\left( \Gamma \right) = \left( {a}_{ij}\right) \) such that each diagonal element \( {a}_{ii} \) equals the number of edges going out of the \( i \) th... | To begin with, assume that \( \Gamma \) itself is a tree. Then show that \( \Gamma \) is a tree if and only if all \( {A}_{ii} \) are equal to 1 . Then attack the general case. | No |
Proposition 10.1 In the linear span of a collection \( \mathbf{u} = \left( {{u}_{1},{u}_{2},\ldots ,{u}_{m}}\right) \) of vectors \( {u}_{i} \in V \), there exists an orthonormal basis \( \mathbf{e} = \left( {{e}_{1},{e}_{2},\ldots ,{e}_{k}}\right) \) such that the transition matrix \( {}^{3}{C}_{eu} = \left( {c}_{ij}\... | Proof Put \( {e}_{1} = {u}_{1}/\left| {u}_{1}\right| \) . Then \( \left| {e}_{1}\right| = 1 \), and \( {e}_{1} \) spans the same subspace as \( {u}_{1} \) . Now assume by induction that the linear span of \( {u}_{1},{u}_{2},\ldots ,{u}_{i} \) possesses the required orthonormal basis \( {e}_{1},{e}_{2},\ldots ,{e}_{\ell... | Yes |
Lemma 10.1 Let \( e = \left( {{e}_{1},{e}_{2},\ldots ,{e}_{n}}\right) \) be an orthonormal basis of a Euclidean space \( V \) . Then \( {\Gamma }_{u} = \mathop{\det }\limits^{2}{C}_{eu} \) for every collection of \( n \) vectors \( \mathbf{u} = \left( {{u}_{1},{u}_{2},\ldots ,{u}_{n}}\right) = \) \( e \cdot {C}_{eu} \)... | Proof Since \( {G}_{u} = {C}_{eu}^{t}{G}_{e}{C}_{eu} = {C}_{eu}^{t}E{C}_{eu} = {C}_{eu}^{t}{C}_{eu} \) and \( \det {C}_{eu} = \det {C}_{eu}^{t} \), we get\n\n\( {\Gamma }_{u} = \det {G}_{u} = \mathop{\det }\limits^{2}{C}_{eu}. \) | Yes |
Proposition 10.2 For every collection of vectors \( \mathbf{u} = \left( {{u}_{1},{u}_{2},\ldots ,{u}_{m}}\right) \), the inequality \( {\Gamma }_{u} = \det \left( \left( {{u}_{i},{u}_{j}}\right) \right) \geq 0 \) holds. It is an equality if and only if the vectors \( {u}_{1},{u}_{2},\ldots ,{u}_{m} \) are linearly rela... | Proof Let \( \mathbf{e} = \left( {{e}_{1},{e}_{2},\ldots ,{e}_{k}}\right) \) be an orthonormal basis in the linear span of vectors \( {u}_{1},{u}_{2},\ldots ,{u}_{m} \) . Then \( {G}_{u} = {C}_{eu}^{t}{C}_{eu} \) . If \( \mathbf{u} \) is linearly independent, then \( \mathbf{u} \) is also a basis, \( k = m \), det \( {... | Yes |
Theorem 10.1 (Orthogonal Decomposition) Let \( U \) be a proper nonzero finite-dimensional subspace of an arbitrary \( {}^{6} \) Euclidean space \( V \). Then\n\n\[ V = U \oplus {U}^{ \bot } \]\n\nand for every vector \( v \in V \), there exists a unique vector \( {v}_{U} \in U \) possessing the following equivalent pr... | Proof Let us verify first that properties (2), (3), (4) are equivalent. The equivalence \( \left( 3\right) \Leftrightarrow \left( 4\right) \) is obvious, because for all \( u \in U \), the equalities \( \left( {v, u}\right) = \left( {{v}_{U}, u}\right) \) and \( \left( {v - {v}_{U}, u}\right) = 0 \) mean the same thing... | Yes |
Example 10.3 (Euclidean Volume of a Parallelepiped Revisited) Let \( V \) be a Euclidean vector space of dimension \( \dim V > n \) . Consider a collection of \( n + 1 \) vectors \( v,{u}_{1},{u}_{2},\ldots ,{u}_{n} \in V \) and write \( U \) for the linear span of the last \( n \) vectors \( {u}_{1},{u}_{2},\ldots ,{u... | \n\[
\operatorname{Vol}\left( {v,{u}_{1},{u}_{2},\ldots ,{u}_{n}}\right) = \left| {\det \left( {v,{u}_{1},{u}_{2},\ldots ,{u}_{n}}\right) }\right| = \left| {\det \left( {h,{u}_{1},{u}_{2},\ldots ,{u}_{n}}\right) }\right|
\]
\n\[
= \sqrt{{\Gamma }_{h,{u}_{1},{u}_{2},\ldots ,{u}_{n}}}.
\]
\nExpansion of the Gram determin... | Yes |
For a point \( a \in {\mathbb{A}}^{n} \) and affine subspace \( \Pi \subset {\mathbb{A}}^{n} \) such that \( a \notin \Pi \), Theorem 10.1 says that the minimal distance \( \left| {a, p}\right| \) taken over all \( p \in \Pi \) is achieved at a unique point \( {a}_{\Pi } \in \Pi \), which is determined by the orthogona... | \[ \left| {a,\Pi }\right| \triangleq \left| {a,{a}_{\Pi }}\right| = \mathop{\min }\limits_{{p \in \Pi }}\left| {a, p}\right| \] is called the distance between \( a \) and \( \Pi \) . If \( a \in \Pi \), we put \( \left| {a,\Pi }\right| = 0 \) . To compute \( \left| {a,\Pi }\right| \), let us fix some point \( q \in \Pi... | Yes |
Example 10.5 (Minimal Distance Method) In practice, the distances between points and affine spaces appear often in the context of minimization. For example, let us find the minimum of the integral\n\n\[ \n{\int }_{0}^{1}{f}^{2}\left( t\right) {dt} \n\]\n\nover all monic cubic polynomials \( f\left( t\right) = {t}^{3} +... | Take \( q = {t}^{3} \) as the origin in \( \Pi \) and use \( {u}_{1} = 1,{u}_{2} = t,{u}_{3} = {t}^{2} \) as a basis in the vectorization of \( \Pi \) centered at \( q \) . Then \( v = \overrightarrow{qa} = - {t}^{3} \) . Since the inner products of basis vectors are\n\n\[ \n\left( {{u}_{i},{u}_{j}}\right) = {\int }_{0... | Yes |
Example 10.6 (Equation of a Hyperplane) For every nonzero vector \( a \in V \) and \( d \in \mathbb{R} \), the inhomogeneous linear equation\n\n\[ \left( {a, x}\right) = d \]\n\n(10.23)\nin the unknown vector \( x \in V \) describes an affine hyperplane in \( \mathbb{A}\left( V\right) \) . | This hyperplane is perpendicular to the vector \( a \) and is shifted the distance \( \left| d\right| /\left| a\right| \) from the origin in the direction of the vector \( a \) for \( d > 0 \) and in the opposite direction for \( d < 0 \) . Indeed, the direction vector space of this hyperplane, which is determined by t... | Yes |
Example 10.7 (Equidistant Plane) For two distinct points \( a, b \in \mathbb{A}\left( V\right) \), their equidistant is the locus of all points \( x \in \mathbb{A}\left( V\right) \) such that \( \left| {x, a}\right| = \left| {x, b}\right| \) . The latter condition is equivalent to the equality \( \left( {a - x, a - x}\... | The distribution of parentheses and cancellation of quadratic terms turns it into the linear equation \( 2\left( {b - a, x}\right) = \left( {b, b}\right) - \left( {a, a}\right) \) . We know from Example 10.6 that this equation describes the hyperplane perpendicular to the segment \( \left\lbrack {a, b}\right\rbrack \) ... | Yes |
For every nonzero vector \( v \in V \) and vector \( u \) running through some subspace \( U \subset V \), the angle \( \measuredangle \left( {v, u}\right) \) either equals \( \pi /2 \) for all \( u \in U \) or reaches its minimal value over all \( u \in U \) exactly on the ray of vectors \( \lambda {\pi }_{U}v,\lambda... | \[ \cos \left( {\measuredangle \left( {v, u}\right) }\right) = \frac{\left( v, u\right) }{\left| v\right| \cdot \left| u\right| } = \frac{\left( {\pi }_{U}v, u\right) }{\left| v\right| \cdot \left| u\right| } = \frac{1}{\left| v\right| } \cdot \left( {{\pi }_{U}v, u/\left| u\right| }\right) . \] The Cauchy-Bunyakovsky-... | Yes |
Example 10.9 (Angle Between Hyperplanes) Let \( \dim V = n \geq 2 \) . Then for any two hyperplanes \( {\Pi }_{1},{\Pi }_{2} \subset {\mathbb{A}}^{n} = \mathbb{A}\left( V\right) \) with different direction subspaces \( {W}_{1},{W}_{2} \subset V \) , the codimension \( \operatorname{codim}\left( {{W}_{1} \cap {W}_{2}}\r... | Exercise 10.10 Check this. | No |
Example 10.13 (Proper Isometries of 3-Dimensional Euclidean Space) By Exercise 10.13, a proper isometry \( F \) of 3-dimensional Euclidean space \( V \) is either the identity map \( {\operatorname{Id}}_{V} \) or a composition of reflections \( F = {\sigma }_{b}{\sigma }_{a} \) in two different 2- dimensional planes \(... | As was explained at the very end of Example 10.12, such a composition is the rotation about the line \( {a}^{ \bot } \cap {b}^{ \bot } \), which is perpendicular to both vectors \( a, b \), in the direction from the plane \( {a}^{ \bot } \) to the plane \( {b}^{ \bot } \) by the doubled acute angle between them. Thus, ... | Yes |
Problem 10.19 (Adjoint Linear Maps) Show that for every linear map of Euclidean vector spaces \( F : U \rightarrow W \) there exists a unique linear map \( {}^{26} \) \( {F}^{ \vee } : W \rightarrow V \) such that \( \left( {{F}^{ \vee }w, u}\right) = \left( {w,{Fu}}\right) \) for all \( w \in W \) and \( u \in U \) . | Check that \( {\left( {F}_{1} \circ {F}_{2}\right) }^{ \vee } = {F}_{2}^{ \vee } \circ {F}_{1}^{ \vee },\ker {F}^{ \vee } = {\left( \operatorname{im}F\right) }^{ \bot },\operatorname{im}{F}^{ \vee } = {\left( \ker F\right) }^{ \bot } \) . For bases \( \mathbf{u} = \left( {{u}_{1},{u}_{2},\ldots ,{u}_{n}}\right) ,\mathb... | No |
Example 11.1 (The Projective Line) The projective line \( {\mathbb{P}}_{1} = \mathbb{P}\left( {\mathbb{k}}^{2}\right) \) is covered by two affine charts \( {U}_{0} = {U}_{{x}_{0}},{U}_{1} = {U}_{{x}_{1}} \), that is, by two affine lines defined by the equations \( {x}_{0} = 1 \) and \( {x}_{1} = 1 \) (see Fig. 11.2). T... | For \( \mathbb{k} = \mathbb{R} \), the result of such gluing can be identified with a circle of diameter 1 glued from two diametrically opposite tangent lines mapped onto the circle via central projections from the opposite points of tangency (see Fig. 11.3). Similarly, for \( \mathbb{k} = \mathbb{C} \), two copies of ... | Yes |
Proposition 11.1 For an infinite ground field \( \mathbb{k} \), the homomorphism (11.2) is injective. If \( \mathbb{k} \) is finite and \( \dim V < \infty \), then the homomorphism (11.2) is surjective and has nonzero kernel. | Proof The first statement was already established in Exercise 9.6 on p. 213. For \( \mathbb{k} = {\mathbb{F}}_{q} \) and \( V = {\mathbb{F}}_{q}^{n} \), the functions \( V \rightarrow \mathbb{k} \) form a vector space of dimension \( {q}^{n} \) with a basis formed by \( \delta \) -functions \( {\delta }_{p}, p \in {\ma... | No |
Corollary 11.2 Under the assumptions of Proposition 11.2, an isomorphism of \( \mathbb{k} \) -algebras \( \sigma : \mathbb{k}\left\lbrack {{x}_{0},{x}_{1},\ldots ,{x}_{n}}\right\rbrack \simeq S{V}^{ * } \) is well defined by sending each basic monomial \( {x}_{0}^{{m}_{0}}{x}_{1}^{{m}_{1}}\cdots {x}_{n}^{{m}_{n}} \in \... | Proof Since \( \sigma \) bijectively maps a basis to a basis, it is an isomorphism of vector spaces. Since \( \sigma \) respects multiplication of basis vectors, it is a homomorphism of algebras. | Yes |
Example 11.3 (Projective Subspaces) The simplest examples of projective varieties are provided by the projective subspaces \( \mathbb{P}\left( U\right) \subset \mathbb{P}\left( V\right) \) associated with vector subspaces \( U \subset V \) . They are given by systems of homogeneous linear equations \( \xi \left( v\righ... | For example, for every pair of nonproportional vectors \( {}^{7}a, b \in V \), there exists a unique projective line \( \left( {ab}\right) \subset \mathbb{P}\left( V\right) \) passing through both \( a \) and \( b \) . Such a line is the projectivization of the 2-dimensional vector subspace spanned by \( a, b \) . It c... | Yes |
Example 11.4 (Smooth Real Conic) The second-degree curve \( C \subset {\mathbb{P}}_{2} = \mathbb{P}\left( {\mathbb{R}}^{3}\right) \) given by the equation\n\n\[ \n{x}_{0}^{2} + {x}_{1}^{2} = {x}_{2}^{2} \n\]\n\n(11.12)\n\nis called a smooth real conic (Fig. 11.5). | In the standard chart \( {U}_{1} \), where \( {x}_{1} = 1 \), in local affine coordinates \( {t}_{0} = {\left. {x}_{0}\right| }_{{U}_{1}} = {\left. {x}_{0}/{x}_{1},{t}_{2} = {x}_{2}\right| }_{{U}_{1}} = {x}_{2}/{x}_{1} \), equation (11.12) becomes the equation of a hyperbola, \( {t}_{2}^{2} - {t}_{0}^{2} = 1 \) . In th... | Yes |
An unordered collection of \( d \) points (some of which may coincide)\n\n\[ \n{p}_{1},{p}_{2},\ldots ,{p}_{d} \in {\mathbb{P}}_{1},\;{p}_{v} = \left( {{p}_{v,0} : {p}_{v,1}}\right) ,\n\]\n\ncan be viewed as a projective hypersurface, the zero set of the homogeneous degree- \( d \) polynomial\n\n\[ \nf\left( {{x}_{0},{... | By analogy with inhomogeneous polynomials \( f\left( t\right) \in \mathbb{k}\left\lbrack t\right\rbrack \), whose roots are collections of points in \( {\mathbb{A}}_{1} \), we say that the points (11.13) are the projective roots of the homogeneous polynomial (11.14). Note that projective roots are defined only up to pr... | Yes |
Example 11.7 (Projection of a Conic onto a Line) In Example 11.4, we considered the smooth conic \( C \) given in \( {\mathbb{P}}_{2} \) by the homogeneous equation \( {x}_{0}^{2} + {x}_{1}^{2} = {x}_{2}^{2} \) . Let us project \( C \) from the point \( p = \left( {1 : 0 : 1}\right) \in C \) on the line \( L \) given b... | \[ \left( {{t}_{1} : {t}_{2}}\right) = \left( {{q}_{1} : \left( {{q}_{2} - {q}_{0}}\right) }\right) \] (11.19) \[ \left( {{q}_{0} : {q}_{1} : {q}_{2}}\right) = \left( {\left( {{t}_{1}^{2} - {t}_{2}^{2}}\right) : 2{t}_{1}{t}_{2} : \left( {{t}_{1}^{2} + {t}_{2}^{2}}\right) }\right) . \] The projection becomes bijective w... | Yes |
Theorem 11.1 Let \( U \) and \( W \) be vector spaces of the same dimension \( \dim U = \) \( \dim W = n + 1 \) . Given two ordered collections of \( n + 2 \) points \( {p}_{0},{p}_{1},\ldots ,{p}_{n + 1} \in \) \( \mathbb{P}\left( U\right) ,{q}_{0},{q}_{1},\ldots ,{q}_{n + 1} \in \mathbb{P}\left( W\right) \) such that... | Proof Fix nonzero vectors \( {u}_{i},{w}_{i} \) representing points \( {p}_{i},{q}_{i} \) and use \( {u}_{0},{u}_{1},\ldots ,{u}_{n} \) and \( {w}_{0},{w}_{1},\ldots ,{w}_{n} \) as bases for \( U \) and \( W \) . The projective transformation \( \bar{F} \) : \( \mathbb{P}\left( U\right) \rightarrow \mathbb{P}\left( W\r... | Yes |
Example 11.8 (Linear Fractional Transformations of a Line) The group \( {\mathrm{{PGL}}}_{2}\left( \mathbb{k}\right) \) consists of nondegenerate \( 2 \times 2 \) matrices \( A = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \) considered up to proportionality. Such a matrix acts on the coordinate projecti... | \[ t \mapsto \frac{{at} + b}{{ct} + d}. \] This affine notation makes it obvious that proportional matrices produce the same transformation and that for every ordered triple of distinct points \( q, r, s \), there exists a unique linear fractional transformation sending those points to \( \infty ,0,1 \) respectively. I... | Yes |
Example 11.9 (Complete Quadrangle) Associated with a quadruple of points \( a, b \) , \( c, d \in {\mathbb{P}}_{2} \) such that no three of them are collinear is a geometric configuration formed by three pairs of lines spanned by disjoint pairs of points (see Fig. 11.7). | The lines in each pair are called opposite sides of the quadrangle \( {abcd} \) . The points \( a, b, c, d \) are called the vertices of the quadrangle. The opposite sides intersect at a triple of points\n\n\[ x = \left( {ab}\right) \cap \left( {cd}\right) \]\n\n\[ y = \left( {ac}\right) \cap \left( {bd}\right) \]\n\n(... | Yes |
Lemma 12.1 An element \( h = {g}^{k} \) generates the cyclic group \( \langle g\rangle \) of order \( n \) if and only if \( \operatorname{GCD}\left( {k, n}\right) = 1 \) . | Proof Since \( \langle h\rangle \subset \langle g\rangle \), the coincidence \( \langle h\rangle = \langle g\rangle \) is equivalent to the inequality \( \operatorname{ord}h \geq n \) . The equality \( {h}^{m} = {g}^{mk} = e \) holds if and only if \( n \mid {mk} \) . If \( \operatorname{GCD}\left( {n, k}\right) = 1 \)... | Yes |
Theorem 12.1 Every permutation is a composition of disjoint (and therefore commuting) cycles. Such a decomposition is unique up to permutation of the factors. | Proof Since the set \( X = \{ 1,2,\ldots, n\} \) is finite, for every \( x \in X \) and \( g \in {S}_{n} \), there must be repetitions in the infinite sequence \( x\overset{g}{ \mapsto }{gx}\overset{g}{ \mapsto }{g}^{2}x\overset{g}{ \mapsto }{g}^{3}x\overset{g}{ \mapsto }\cdots \) . Since \( g \) is bijective, the left... | Yes |
Exercise 12.8 Write \( {m}_{i} = {m}_{i}\left( \lambda \right) \) for the number of rows of length \( i \) in the Young diagram \( \lambda \) . Therefore, \( 0 \leq {m}_{i} \leq \left| \lambda \right| \) for each \( i \) and \( \mathop{\sum }\limits_{i}i \cdot {m}_{i} = \left| \lambda \right| = n \) . Show that the tot... | \[ \frac{n!}{\mathop{\prod }\limits_{i}{i}^{{m}_{i}} \cdot {m}_{i}!} \] | Yes |
Example 12.3 (How to Compute Order and Sign) The order of a permutation of cyclic type \( \lambda = \left( {{\lambda }_{1},{\lambda }_{2},\ldots ,{\lambda }_{s}}\right) \) is equal to \( \operatorname{LCM}\left( {{\lambda }_{1},{\lambda }_{2},\ldots ,{\lambda }_{s}}\right) \) . For example, the permutation\n\n\[ \left(... | The thread rule \( {}^{6} \) shows that the sign of a cycle of length \( m \) equals \( {\left( -1\right) }^{m - 1} \) . Hence a permutation is even if and only if it has an even number of even-length cycles. | No |
Consider a regular tetrahedron centered at the origin of \( {\mathbb{R}}^{3} \). Since an isometry of \( {\mathbb{R}}^{3} \) is uniquely determined by its action on the vertices of the tetrahedron and this action can be chosen arbitrarily, the complete group of the regular tetrahedron \( {\mathrm{O}}_{\text{tet }} \) i... | Indeed, a rotation of the tetrahedron is uniquely determined by its action on the affine coordinate system formed by a vertex and the outgoing three edges from it. The vertex can be mapped to any one of four vertices, whereupon we have exactly three possibilities for an orientation-preserving superposition of edges. Th... | Yes |
Proposition 12.1 All nonempty fibers of a group homomorphism \( \varphi : {G}_{1} \rightarrow {G}_{2} \) are in bijection with \( \ker \varphi \) . Namely, \( {\varphi }^{-1}\left( {\varphi \left( g\right) }\right) = g \cdot \left( {\ker \varphi }\right) = \left( {\ker \varphi }\right) \cdot g \) for every \( g \in {G}... | Proof If \( \varphi \left( t\right) = \varphi \left( g\right) \), then \( \varphi \left( {t{g}^{-1}}\right) = \varphi \left( t\right) \varphi {\left( g\right) }^{-1} = e \) and \( \varphi \left( {{g}^{-1}t}\right) = \) \( \varphi {\left( g\right) }^{-1}\varphi \left( t\right) = e \) . Hence, \( t{g}^{-1} \in \ker \varp... | Yes |
Example 12.7 (Sign Homomorphism) In Corollary 9.1 on p.209, we constructed the sign homomorphism sgn : \( {S}_{n} \rightarrow \{ \pm 1\} \) from the symmetric group to the multiplicative group of signs. | Its kernel is the group of even permutations \( {A}_{n} = \) ker sgn of order \( \left| {A}_{n}\right| = n!/2 \). | Yes |
Example 12.8 (Determinant and Finite Linear Groups) In Sect. 9.3.2 on p. 214 we constructed the determinant homomorphism\n\n\\[ \n\\det : \\mathrm{{GL}}\\left( V\\right) \\rightarrow {\\mathbb{k}}^{ * },\\;F \\mapsto \\det F, \n\\]\n\n(12.5)\n\nfrom the general linear group \\( \\mathrm{{GL}}\\left( V\\right) \\) of a ... | Since the determinant homomorphism (12.5) is surjective, \\( {}^{14} \\) the special linear group has order \\( \\left| {{\\mathrm{{SL}}}_{n}\\left( {\\mathbb{F}}_{q}\\right) }\\right| = \\left| {{\\mathrm{{GL}}}_{n}\\left( {\\mathbb{F}}_{q}\\right) }\\right| /\\left| {\\mathbb{k}}^{ * }\\right| = \\left( {{q}^{n} - 1}... | Yes |
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