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Unformatted text preview: MA 265 LECTURE NOTES: FRIDAY, APRIL 11 Eigenvalues and Eigenvectors Review. Let ( V, + , ) be a real vector space of dimension n . Recall that a linear operator is a function L : V V such that L preserves vector addition: For all u , v V , we have L ( u + v ) = L ( u ) + L ( v ). L preserves scalar multiplication: For all scalars c R and vectors u V , we have L ( c u ) = cL ( u ). In practice, we will choose V = R n . Recall that there is an n n matrix A , the standard matrix representing L , such that L ( x ) = A x . We would like to find all vectors x V such that L ( x ) is parallel to x , i.e., L ( x ) = x for some scalar . Example. Consider V = R 2 . As in the last lecture, fix a positive scalar r and consider the linear operator L : R 2 R 2 defined as L ( x ) = A x in terms of the 2 2 matrix A = r r = L x y = r x r y . Recall that this is either a dilation (if r > 1) or a contraction (if 0 < r < 1). Hence L ( x ) = x for all x R 2 where = r . Note that A is a scalar matrix. Example. As another example along similar lines, consider the 2 2 matrix A = 1 1 = L x y = x y . Recall that this is the reflection about the xaxis. In particular, there is no scalar so that L ( x ) = x for all vectors x V . We will show, however, that there are subspaces W of V = R 2 where we can find such a scalar....
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This note was uploaded on 03/11/2010 for the course MA 261A 0026100 taught by Professor ... during the Spring '10 term at Purdue University Calumet.
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