david 02
1
6. Let L : IR 2 ^ IR 2 be the linear transformation on IR 2 that rotates each vector in IR 2 counterclockwise by an angle of 9. Find the matrix representing L with respect to the standard basis for R 2 . [25 pts.] 7. Prove that (Asm6 + Bcos(j)) 2 < 2{A 2 + B 2 ). for all real numbers A, B, 9, and <$> . Hint: Use the Cauchy-Schwarz inequality. [25 pts.] Bonus [5 pts.]: When is equality attained in no. 7? (Give all the conditions for A,B,9, a n d </>). evb/mal22rodenfinals
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david LT 2
Transcript of david 02
6. Let L : IR 2 ^ IR 2 be the linear transformation on IR 2 that rotates each vector in IR 2
counterclockwise by an angle of 9. F i n d the matr ix representing L w i t h respect to the standard basis for R 2 . [25 pts.]
7. Prove that
( A s m 6 + B c o s ( j ) ) 2 < 2 { A 2 + B 2 ) .
for all real numbers A , B , 9, and <$>. Hint : Use the Cauchy-Schwarz inequality. [25 pts.]
Bonus [5 pts.]: W h e n is equality attained in no. 7? (Give a l l the cond i t i ons for A , B , 9 , and </>).
evb /mal22rodenf ina ls
