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