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| | | 45 yr old Painting Positions Worker Carmouche from Saint-Hubert, loves to spend time caravaning, property developers in singapore and frisbee. In the last year has made a trip to Central Sikhote-Alin.<br><br>Here is my page :: [http://www.geopolle.com/users/chasebrobst292494462 www.geopolle.com] |
| In [[mathematics]], in [[linear algebra]], a '''cyclic subspace''' is a certain special [[Linear subspace|subspace]] of a [[dimension|finite-dimensional]] [[vector space]] associated with a vector in the vector space and a [[linear map|linear transformation]] of the vector space. The cyclic subspace associated with a vector ''v'' in a vector space ''V'' and a linear transformation ''T'' of ''V'' is called the ''' ''T''-cyclic subspace generated by ''v'''''. The concept of a cyclic subspace is a basic component in the formulation of the cyclic decomposition theorem in linear algebra.
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| ==Definition==
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| Let <math>T:V\rightarrow V</math> be a linear transformation of a vector space <math>V</math> and let <math> v</math> be a vector in <math>V</math>. The <math>T</math>-cyclic subspace of <math>V</math> generated by <math>v</math> is the subspace <math>W</math> of <math>V</math> generated by the set of vectors <math>\{ v, T(v), T^2(v), \ldots, T^r(v), \ldots\}</math>. This subspace is denoted by <math> Z(v;T)</math>. If <math>V=Z(v;T)</math>, then <math>v</math> is called a '''cyclic vector''' for <math>T</math>.<ref name=LA>{{cite book|last=Kenneth Hoffmann, Ray Kunze|title=Linear Algebra (Second Edition)|year=1971|publisher=Prentice-Hall|page=227}}</ref>
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| There is another equivalent definition of cyclic spaces. Let <math>T:V\rightarrow V</math> be a linear transformation of a finite dimensional vector space over a [[field (mathematics)|field]] <math>F</math> and <math>v</math> be a vector in <math>V</math>. The set of all vectors of the form <math>g(T)v</math>, where <math>g(x)</math> is a [[polynomial]] in the [[Ring (mathematics)|ring]] <math>F[x]</math> of all polynomials in <math>x</math> over <math>F</math>, is the <math>T</math>-cyclic subspace generated by <math>v</math>.<ref name="LA"/>
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| ===Examples===
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| # For any vector space <math>V</math> and any linear operator <math>T</math> on <math>V</math>, the <math>T</math>-cyclic subspace generated by the zero vector is the zero-subspace of <math>V</math>.
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| # If <math>I</math> is the [[identity operator]] then every <math>I</math>-cyclic subspace is one-dimensional.
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| # <math>Z(v;T)</math> is one-dimensional if and only if <math>v</math> is a [[characteristic vector]] of <math>T</math>.
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| # Let <math>V</math> be the two-dimensional vector space and let <math>T</math> be the linear operator on <math>V</math> represented by the matrix <math>\begin{bmatrix} 0&1\\ 0&0\end{bmatrix}</math> relative to the standard ordered basis of <math>V</math>. Let <math>v=\begin{bmatrix} 0 \\ 1 \end{bmatrix}</math>. Then <math> Tv = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \quad T^2v=0, \ldots, T^rv=0, \ldots </math>. Therefore <math>\{ v, T(v), T^2(v), \ldots, T^r(v), \ldots\} = \left\{ \begin{bmatrix} 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right\}</math> and so <math>Z(v;T)=V</math>. Thus <math>v</math> is a cyclic vector for <math>T</math>.
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| ==Companion matrix==
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| Let <math>T:V\rightarrow V </math> be a linear transformation of a <math>n-</math>dimensional vector space <math>V</math> over a field <math>F</math> and <math>v</math> be a cyclic vector for <math>T</math>. Then the vectors
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| ::<math>B=\{v_1=v, v_2=Tv, v_3=T^2v, \ldots v_n = T^{n-1}v\}</math> | |
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| form an ordered basis for <math>V</math>. Let the characteristic polynomial for <math>T</math> be
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| ::<math> p(x)=c_0+c_1x+c_2x^2+\cdots + c_{n-1}x^{n-1}+x^n</math>.
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| Then
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| ::<math>
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| \begin{align}
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| Tv_1 & = v_2\\
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| Tv_2 & = v_3\\
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| Tv_3 & = v_4\\
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| \vdots & \\
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| Tv_{n-1} & = v_n\\
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| Tv_n &= -c_0v_1 -c_1v_2 - \cdots c_{n-1}v_n\\
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| \end{align}
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| </math>
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| Therefore, relative to the ordered basis <math>B</math>, the operator <math>T</math> is represented by the matrix
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| ::<math>
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| \begin{bmatrix}
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| 0 & 0 & 0 & \cdots & 0 & -c_0 \\
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| 1 & 0 & 0 & \ldots & 0 & -c_1 \\
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| 0 & 1 & 0 & \ldots & 0 & -c_2 \\
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| \vdots & & & & & \\
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| 0 & 0 & 0 & \ldots & 1 & -c_{n-1} \\
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| \end{bmatrix}
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| </math>
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| This matrix is called the ''companion matrix'' of the polynomial <math>p(x)</math>.<ref name="LA"/>
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| ==See also==
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| * [[Companion matrix]]
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| * [[Cyclic decomposition theorem]]
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| ==External links==
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| * PlanetMath: [http://planetmath.org/cyclicsubspace cyclic subspace]
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| ==References==
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| {{reflist}}
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| [[Category:Linear algebra]]
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45 yr old Painting Positions Worker Carmouche from Saint-Hubert, loves to spend time caravaning, property developers in singapore and frisbee. In the last year has made a trip to Central Sikhote-Alin.
Here is my page :: www.geopolle.com