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invertingmatrices - Inverting Matrices Math 108A John...

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Inverting Matrices Math 108A: May 12, 2010 John Douglas Moore Recall that any linear transformation T : R n R n can be represented in terms of the standard basis by an n × n matrix A = a 11 a 12 · · · a 1 n a 21 a 22 · · · a 2 n · · · · · · a n 1 a n 2 · · · a nn . In other words, the linear transformation can be represented in terms of the vector x = x 1 x 2 · x n as T ( x ) = A x . We can think of the linear transformation as taking x to y = y 1 y 2 · y n = a 11 a 12 · · · a 1 n a 21 a 22 · · · a 2 n · · · · · · a n 1 a n 2 · · · a nn x 1 x 2 · x n . If the linear transformation is one to one and onto, it possesses an inverse map T - 1 , which is also a linear transformation, and is represented by a matrix B = b 11 b 12 · · · b 1 n b 21 b 22 · · · b 2 n · · · · · · b n 1 b n 2 · · · b nn ,
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