Let T R3 rightarrow R3 such that T 1 0 1 1 1 4 T 0 1 1 2 0

Let T: R^3 rightarrow R^3 such that T (-1 0 1) = (-1 1 -4), T (0 1 -1) = (2 0 2) and T (1 1 -1) = (3 0 4). Determine T (-3 0 2). Determine T (x y z).

Denote transpose by: \'

(a)

Let, a, b,c so that:

a(-1,0,1)\'+b(0,1,-1)\'+c(1,1,-1)\'=(3,0,2)\'

This gives:

b=-c

-a+c=3

a-b-c=2 , hence, a=2

c=5, b=-5

Let,

T(-3,0,2)\'=2T(-1,0,1)\'-5T(0,1,-1)\'+5T(1,1,-1)\'=2(-1,1,-4)\'-5(2,0,2)+5(3,0,4)=(3,2,2)

(b)

Let, X=(-1,0,1)\',Y=(0,1,-1)\',Z=(1,1,-1)\'

e1=Z-Y=(1,0,0)\'

e3=X+e1=X+Z-Y=(0,0,1)\'

e2=Z+e3-e1=Z+X+Z-Y-(Z-Y)=X+Z=(0,1,0)\'

So, (x,y,z)\'=xe1+ye2+ze3

So,

T(x,y,z)\'=xT(e1)+yT(e2)+zT(e3)=x(T(X)-T(Y))+y(T(X)+T(Z))+z(T(X)+T(Z)-T(Y))

T(x,y,z)\'=(x+y+z)T(X)+(-x-z)T(Y)+(y+z)T(Z)