Suppose T 32 is a linear transformation Let U and V be the v

Suppose T: ?³??² is a linear transformation. Let U and V be the vectors given below, and suppose that T(U) and T(V) are as given. Find T(3U+2V).

U =

1

2

2

V =

?2

?3

?3

T(U) =

?2

?1

T(V) =

3

1

T(3U+2V) =

[?]

[?]

Solution

Since T is a linear transformation, it is closed under vector addition and scalar multiplication so that.

T(3U+2V) = T( 3U) +T( 2V)= 3 T(U) +2T(V) = 3(-2, 1)^T +2( 3, 1)^T = (-6, 3)^T + (6, 2)^T = ( 0, 5)^T.