Unformatted text preview: ) are linearly dependent. (f) Show that if L ( −→ ı ) 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 ( −→ ı ) 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 ( −→ ı ) and L ( −→ ), ±rst adjust the angle, then get the lengths right, and ±nally rotate into position.)...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Spring '08 term at University of Florida.
- Spring '08