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Classification of Conic Sections

  • Thomas Banchoff
  • John Wermer
Part of the Undergraduate Texts in Mathematics book series (UTM)

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,
$$ m(A)\left( {\begin{array}{*{20}{c}} {{x_1}}\\ {{y_1}} \end{array}\;\;\;\begin{array}{*{20}{c}} {{x_2}}\\ {{x_1}} \end{array}} \right)\;{\rm{ = }}\;\left( {\begin{array}{*{20}{c}} {{{x'}_1}}\\ {{{y'}_1}} \end{array}\;\;\;\begin{array}{*{20}{c}} {{{x'}_2}}\\ {{{y'}_2}} \end{array}} \right)$$
(1)
.

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Copyright information

© Springer-Verlag New York, Inc. 1992

Authors and Affiliations

  • Thomas Banchoff
    • 1
  • John Wermer
    • 1
  1. 1.Department of MathematicsBrown UniversityProvidenceUSA

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