# 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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© Springer-Verlag New York, Inc. 1992

## Authors and Affiliations

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