14-240/Tutorial-November11: Difference between revisions
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Let <math>V</math> be a finite dimensional vector space over a field <math>F</math> |
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 representation''' of <math>v</math> is the column vector \begin{pmatrix}c_1\\c_2\\c_3\\.\\c_n\end{pmatrix}</math>. |
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Let <math>W</math> be a finite dimensional vector space over the same field <math>F</math>, <math>B = {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 the matrix representation of <math>T</math> in the ordered bases <math>B, K</math> is the matrix |
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Revision as of 18:42, 29 November 2014
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Boris
Coordinate and Matrix Representation Problems
Recall:
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 representation of [math]\displaystyle{ v }[/math] is the column vector \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], [math]\displaystyle{ B = {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 the matrix representation of [math]\displaystyle{ T }[/math] in the ordered bases [math]\displaystyle{ B, K }[/math] is the matrix
