Homework Sourse1 If T is a linear transformation and Tv1 121 and Tv2 201
1. If T is a linear transformation and T(v1) = (1,2,1) and T(v2) = (2,0,-1) what is T(4v1 - 3v2)
2. If T is a linear transformation and T(v1) = (2, 2/3, 1) and T(v2) = (1,-1,0) what is T(3v1 - 4v2)
Solution
1) Given that
T(v1) = ( 1 , 2 , 1 ) , T(v2) = ( 2 , 0 , -1 )
T( 4v1 - 3v2 ) = T( 4v1) - T( 3v2 ) [ since,T(u - v ) = T(u) - T(v) ]
= 4T(v1) - 3T(v2) [ T( cv) = c T(v) ,where c = constant ]
= 4 ( 1 , 2 , 1 ) - 3( 2 , 0 , -1 )
= ( 4 , 8 , 4 ) - ( 6 , 0 , -3 )
= ( -2 , 8 , 7 )
Therefore,
T( 4v1 - 3v2 ) = ( -2 , 8 , 7 )
2 ) Given that
T(v1) = ( 2 , 2/3 , 1 ) , T(v2) = ( 1 , -1 , 0 )
T( 3v1 - 4v2 ) = T( 3v1) - T( 4v2 ) [ since,T(u - v ) = T(u) - T(v) ]
= 3T(v1) - 4T(v2) [ T( cv) = c T(v) ,where c = constant ]
= 3 ( 2 , 2/3 , 1 ) - 4 ( 1 , -1 , 0 )
= ( 6 , 2 , 3 ) - ( 4 , -4 , 0 )
= ( 2 , 6 , 3 )
Therefore,
T( 3v1 - 4v2 ) = ( 2 , 6 , 3 )
