14-240/Tutorial-November11
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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 [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].
