Abstract
We can use matrix multiplication to keep track of the action of a transformation A on a pair of vectors \( \left( {\begin{array}{*{20}{c}} {{x_1}}\\ {{y_1}} \end{array}} \right)\;{\rm{and}}\;\left( {\begin{array}{*{20}{c}} {{x_2}}\\ {{y_2}} \end{array}} \right){\rm{.}} \) Let m(A) be the matrix of A, and consider the matrix \( \left( {\begin{array}{*{20}{c}} {{x_1}}\\ {{y_1}} \end{array}\;\;\;\begin{array}{*{20}{c}} {{x_2}}\\ {{y_2}} \end{array}} \right) \) whose columns are \( \left( {\begin{array}{*{20}{c}} {{x_1}}\\ {{y_1}} \end{array}} \right)\;{\rm{and}}\;\left( {\begin{array}{*{20}{c}} {{x_2}}\\ {{y_2}} \end{array}} \right){\rm{.}} \) If we set \( \left( {\begin{array}{*{20}{c}} {{{x'}_1}}\\ {{{y'}_1}} \end{array}} \right)\;{\rm{ = }}\;A\left( {\begin{array}{*{20}{c}} {{x_1}}\\ {{y_1}} \end{array}} \right)\;{\rm{and}}\;\left( {\begin{array}{*{20}{c}} {{{x'}_2}}\\ {{{y'}_2}} \end{array}} \right)\;{\rm{ = }}\;A\left( {\begin{array}{*{20}{c}} {{x_2}}\\ {{y_2}} \end{array}} \right), \) then, as we shall prove,
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© 1992 Springer-Verlag New York, Inc.
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Banchoff, T., Wermer, J. (1992). Classification of Conic Sections. In: Linear Algebra Through Geometry. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4390-8_9
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DOI: https://doi.org/10.1007/978-1-4612-4390-8_9
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