Let

Question

Let

L: R2 ¨ R2
 be a linear transformation for which we know that

L(1 1)= [-2 1]
L(-1 1)=[3 2]

a)what is L(-1 5)

b)what is L(u v)?

Qu · Accepted Answer

Quest qe jupoiur

MathMate · Answer

Use the properties of linear transformations:
1. T(A+B)=T(A)+T(B)
2. T(kA)=kT(A)

then
let A=[1,1], B=[-1,1]
and
T(A+B)=T(0,2)=[-2,1]+[3,2]=[1,3]
=>
T(0,1)=[1/2, 3/2]=P
and
T(A-B)=T(2,0)=[-2,1]-[3,2]=[-5,-1]
=>
T(1,0)=[-5/2, -1/2]=Q

So
(a)
L(-1,5)=-P+5Q
=-[1/2,3/2]+5[-5/2,-1/2]
=[-13, -4]

(b)
L(u,v)=uP+vQ=...