**Unformatted text preview: **β )-cos(2 α ) sin(2 β ) cos(2 α ) cos(2 β ) + sin(2 α ) sin(2 β ) ² = ± cos (2( α-β ))-sin (2( α-β )) sin (2( α-β )) cos (2( α-β )) ² . Thus R α ◦ R β is a rotation by 2( α-β ). Similarly, R β ◦ R α is a rotation by 2( β-α ). These rotations are equal when α and β are congruent modulo π/ 2. 2. [4 marks] Let K : R m → R n and L : R n → R m be linear transformations, and let m < n . Prove that the reduced row-echelon form of the standard matrix of K ◦ L is not an identity matrix. Let K and L have standard matrices A and B , respectively. AB , the standard matrix of K ◦ L , is m × m . Now the reduced row-echelon form of AB is the m × m identity matrix if, and only if, Nul( AB ) = { } . Since m < n , B has at least one free variable. Hence there exists a nonzero vector x ∈ R m such that B x = , and AB x = . Thus Nul( AB ) 6 = { } ....

View
Full Document

- Spring '09
- FokShuen
- Linear Algebra, Algebra, matlab, Addition, Vector Space, Linear map, Identity function, Rθ