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Unformatted text preview: ) are linearly dependent. (f) Show that if L ( ) and L ( ) are dependent, then there is some nonzero vector sent to 0 by L . (g) Use this to prove that the following are equivalent: i. the determinant of [ L ] is nonzero; ii. L ( ) and L ( ) are linearly independent; iii. L is onto; iv. L is one-to-one. (h) L is invertible if there exists another map F : R 2 R 2 such that L ( F ( x,y )) = ( x,y ) = F ( L ( x,y )). Show that if F exists it must be linear. 7. Show that every invertible linear map L : R 2 R 2 can be expressed as a composition of the kinds of mappings described in Exercise 5 . ( Hint: Given the desired images L ( ) and L ( ), rst adjust the angle, then get the lengths right, and nally rotate into position.)...
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- Spring '08