50 lines
598 B
TeX
50 lines
598 B
TeX
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Linear Algebra SP3
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\bigskip
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{\bf 2a.} Find the standard matrix $[T]_{std}$ for the following
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linear transformation.
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$$
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T:R^2\rightarrow R^2
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\quad
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\hbox{such that}
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\quad
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T\pmatrix{a\cr b}
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=
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\pmatrix{2a+b\cr a-b}
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$$
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\bigskip
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To solve, apply the transformation to standard
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basis vectors.
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$$
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T\pmatrix{1\cr0}
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=
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\pmatrix{2\cr1},
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\quad
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T\pmatrix{0\cr1}
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=
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\pmatrix{1\cr-1}
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$$
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The results become the columns of the standard matrix.
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$$[T]_{std}=\pmatrix{2&1\cr1&-1}$$
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Multiply to check.
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$$
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\pmatrix{2&1\cr1&-1}
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\pmatrix{a\cr b}
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=
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\pmatrix{2a+b\cr a-b}
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$$
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{\it See Unit 12.}
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\end
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