Titlebook: Linear Algebra Through Geometry; Thomas Banchoff,John Wermer Textbook 1992Latest edition Springer-Verlag New York, Inc. 1992 Eigenvalue.Li

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Products of Linear Transformations,Let . and . be two linear transformations. We define the transformation . which consists of . followed by ., i.e., if . is any vector.We write . = . and we call ..

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Determinants,Let . be a linear transformation with matrix .. The quantity.is called the determinant of the matrix . and is denoted.Expressed in these terms, Theorem 2.4 states that . has an inverse if and only if . We shall see that the determinant gives us further information about the behavior of ..

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Vector Geometry in 3-Space,Just as in the plane, we may use vectors to express the analytic geometry of 3-dimensional space.

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Sums and Products of Linear Transformations,If . and . are linear transformations, then we may define a new transformation . + . by the condition.Then by definition, (. + .)(. + .) = .(. + .) + .(. + .), and since . and . are linear transformations, this equals .(.) + .(.) + .(.) + .(.) = .(.) + .(.) + .(.) + .(.) = (. + .)(.) + (. + .)(.). Thus for every pair ., we have

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