Okay, so we have a linear transformation T: R^3 --> R^2. We are then searching for a matrix A, since the transformation is defined by T(x) = Ax. Since we know that x is a column vector (3 by 1) and that the product of Ax is a 2 by 1 column vector we're searching for a matrix A which is a 2 by 3 matrix.

A =

[a b c]

[d e f]

We also have that

T([1;0;1]) = [0;1]

T([0;-1;1]) = [1;0]

T([1;-1;0]) = [0;0]

Using the fact that T(x) = Ax we can get a system of equation for the three given transformations. Multiply the yet unknown matrix A with the entries a,b,c,d,e,f. It will look like:

a*1+0*b+1*c = 0

d*1+e*0+1*f = 1

a*0+b*-1+c*1 = 1

d*0+e*-1+f*1 = 0

a*1+b*-1+c*0 = 0

d*1+e*-1+f*0 = 0

You can write this as an augmented matrix with 6 unknowns

[1 0 1 0 0 0 0]

[0 0 0 1 0 1 1]

[0 -1 1 0 0 0 1]

[0 0 0 0 -1 1 0]

[1 -1 0 0 0 0 0]

[0 0 0 1 -1 0 0]

Which is row equivalent to

[1 0 0 0 0 0 -1/2]

[0 1 0 0 0 0 -1/2]

[0 0 1 0 0 0 1/2]

[0 0 0 1 0 0 1/2]

[0 0 0 0 1 0 1/2]

[0 0 0 0 0 1 1/2]

We then have

a = -1/2

b = -1/2

c = 1/2

d = 1/2

e = 1/2

f = 1/2

A =

[-1/2 -1/2 1/2]

[ 1/2 1/2 1/2]

Multiply this with the columnvector [x1;x2;x3] and you have your given transformation