Finding a basis for the subspace U with constraint a+b=c+d in R4

[stunner5000pt]

Find a basis and calculate the dimension of the following subspaces of R4
U = [a b c d] | a+b=c+d in R

now a basis is supposed to be linear independent
but that conditon on a,b,c,d makes this a bit strange

can i just say d = a+b - c and then pick
a basis as (1,0,0,1),(0,1,0,1),(0,0,1,-1)

text says this however
(-1,1,0,0),(1,0,1,0),(-1,0,0,1)

please help
i am currently studying for my exam i would like to understand everything!

your help is greatly appreciated!

 

[0rthodontist]

It looks like your text just chose to write a = d + c - b. I'm sure you know there can be more than one basis.