not sure if this is classed as advanced algebra but here goes.
I need to find the matrix of t and I am confused as to which is the correct method to use.
The basis for the domain is E = {(1,0,0),(1,1,0),(0,1,1)} and the basis for the codomain is the standard basis of F = {(1,0,0),(0,1,0),(0,0,1)}
t : R^3 → R^3
(x,y,z) → (y - z, x + z, x + y)
This is what I have worked out so far:
I have found the images of the vectors in the domain with basis E = {(1,0,0),(1,1,0),(0,1,1)} to be
t(1,0,0) = (0,1,1)
t(1,1,0) = (1,1,2)
t(0,1,1) = (1,1,1)
as t(x,y,z) = (y - z, x + z, x + y)
and the F - coordinates of each of the image vectors, where F = {(1,0,0),(0,1,0),(0,0,1)}
t(1,0,0) = (0,1,1)F
t(1,1,0) = (1,1,2)F
t(0,1,1) = (1,1,1)F
so is the matrix of t with respect to the bases of E and F
0 1 1
1 1 2
1 1 1 (sorry, do not know how to do big brackets)
OR...........
do I need to use the below method for finding the matrix?
t(1,0,0) = (0,1,1) = (a,b,c)F
(a,b,c)F = a(1,0,0) + b(0,1,0), c(0,0,1) = (a,b,c)
a=1
b=0
c=0
t(1,1,0) = (1,1,2) = (d,e,f)F
(d,e,f)F = d(1,0,0) + e(0,1,0) + f(0,0,1) = (d,e,f)
d=1
e=1
f=0
t(0,1,1) = (1,1,1) = (g,h,i)F
(g,h,i)F = g(1,0,0) + h(0,1,0) + i(0,0,1) = (g,h,i)
g=0
h=1
i=1
and this gives the matrix of:
1 1 0
0 1 1
0 0 1
I think it is my first choice but need someone to clarify. I'm just getting to grips with the terminology so please be gentle with your answer![]()