14-240/Tutorial-November11: Difference between revisions
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==Boris== |
==Boris== |
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====Useful Definitions==== |
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====Coordinate and Matrix Representation Problems==== |
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| ⚫ | Let <math>V</math> be a finite dimensional vector space over a field <math>F</math>, <math>B = \{v_1, v_2, v_3, ..., v_n\}</math> be an ordered basis of <math>V</math> and <math>v \in V</math>. Then <math>v = \displaystyle\sum_{i=1}^{n} c_iv_i</math> where <math>c_i \in F</math>. Then the '''coordinate vector''' of <math>v</math> relative to <math>B</math> is the column vector <math> \begin{pmatrix}c_1\\c_2\\c_3\\...\\c_n\end{pmatrix}</math>. |
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Recall: |
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| ⚫ | Let <math>V</math> be a finite dimensional vector space over a field <math>F</math> |
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Let <math>W</math> be a finite dimensional vector space over the same field <math>F</math> and <math>K = \{v_1, v_2, v_3, ..., v_m\}</math> be an ordered basis of <math>W</math>. Define a linear transformation <math>T:V \to W</math>. Then <math>T(v_j) = \displaystyle\sum_{i=1}^{m} c_{ij}T(v_j)</math> where <math>c_{ij} \in F</math>. Then the '''matrix representation''' of <math>T</math> in the ordered bases <math>B, K</math> is the matrix <math>\begin{pmatrix}c_{11} & c_{12} & c_{13} & ... & c_{1n}\\c_{21} & c_{22} & c_{23} & ... & c_{2n}\\c_{31} & c_{32} & c_{33} & ... & c_{3n}\\c_{21} & c_{22} & c_{23} & ... & c_{2n}\\... & ... & ... & ...& ... \\c_{m1} & c_{m2} & c_{m3} & ... & c_{mn}\end{pmatrix}</math>. |
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====Boris's Problems==== |
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Let <math>B</math> be the standard ordered basis of <math>P_n(F)</math> and <math>K</math> be the standard ordered basis of <math>F</math>. |
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'''Q1'''. What is the '''coordinate vector''' of <math>x^2 + x^5</math> relative to <math>B</math>? |
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'''Q2'''. Let <math>T:P_n \to F</math> be a linear transformation that is defined by <math>T(f(x)) = f(0)</math>. What is the '''matrix representation''' of <math>T</math> in <math>B, K</math>? |
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==Nikita== |
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Latest revision as of 17:35, 30 November 2014
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Boris
Useful Definitions
Let [math]\displaystyle{ V }[/math] be a finite dimensional vector space over a field [math]\displaystyle{ F }[/math], [math]\displaystyle{ B = \{v_1, v_2, v_3, ..., v_n\} }[/math] be an ordered basis of [math]\displaystyle{ V }[/math] and [math]\displaystyle{ v \in V }[/math]. Then [math]\displaystyle{ v = \displaystyle\sum_{i=1}^{n} c_iv_i }[/math] where [math]\displaystyle{ c_i \in F }[/math]. Then the coordinate vector of [math]\displaystyle{ v }[/math] relative to [math]\displaystyle{ B }[/math] is the column vector [math]\displaystyle{ \begin{pmatrix}c_1\\c_2\\c_3\\...\\c_n\end{pmatrix} }[/math].
Let [math]\displaystyle{ W }[/math] be a finite dimensional vector space over the same field [math]\displaystyle{ F }[/math] and [math]\displaystyle{ K = \{v_1, v_2, v_3, ..., v_m\} }[/math] be an ordered basis of [math]\displaystyle{ W }[/math]. Define a linear transformation [math]\displaystyle{ T:V \to W }[/math]. Then [math]\displaystyle{ T(v_j) = \displaystyle\sum_{i=1}^{m} c_{ij}T(v_j) }[/math] where [math]\displaystyle{ c_{ij} \in F }[/math]. Then the matrix representation of [math]\displaystyle{ T }[/math] in the ordered bases [math]\displaystyle{ B, K }[/math] is the matrix [math]\displaystyle{ \begin{pmatrix}c_{11} & c_{12} & c_{13} & ... & c_{1n}\\c_{21} & c_{22} & c_{23} & ... & c_{2n}\\c_{31} & c_{32} & c_{33} & ... & c_{3n}\\c_{21} & c_{22} & c_{23} & ... & c_{2n}\\... & ... & ... & ...& ... \\c_{m1} & c_{m2} & c_{m3} & ... & c_{mn}\end{pmatrix} }[/math].
Boris's Problems
Let [math]\displaystyle{ B }[/math] be the standard ordered basis of [math]\displaystyle{ P_n(F) }[/math] and [math]\displaystyle{ K }[/math] be the standard ordered basis of [math]\displaystyle{ F }[/math].
Q1. What is the coordinate vector of [math]\displaystyle{ x^2 + x^5 }[/math] relative to [math]\displaystyle{ B }[/math]?
Q2. Let [math]\displaystyle{ T:P_n \to F }[/math] be a linear transformation that is defined by [math]\displaystyle{ T(f(x)) = f(0) }[/math]. What is the matrix representation of [math]\displaystyle{ T }[/math] in [math]\displaystyle{ B, K }[/math]?
