Are the two spans equal?

Let $a=x+y, b=y+z, c=z+x.$

Then $$pmatrix{a\b\c} = pmatrix{1&1&0\0&1&1\1&0&1}pmatrix{x\y\z}$$ and $$ pmatrix{x\y\z}= frac{1}{2}pmatrix{1&-1&1\1&1&-1\-1&1&1}pmatrix{a\b\c}$$

Since there is an invertible transformation from one set of vectors to the other, the spans are equivalent.

Assume W := v ∈ span{x+y, y+z, z+x} then

v = a(x+y) + b(y+z) + c(z+x)

= (a+c)x + (a+b)y + (b+c)z

so that if v ∈ W then v ∈ U, i.e. W ⊆ U. Assume v ∈ U then

v = ax + by + cz

= ((a+b−c)/2)(x+y) + ((b+c−a)/2)(y+z) + ((a−b+c)/2)(z+x) ∈ W

i.e. U ⊆ W.

$x = frac{1}{2}(x+y)-frac{1}{2}(y+z)+frac{1}{2}(z+x)$

$y = frac{1}{2}(x+y) -frac{1}{2}(z+x) + frac{1}{2}(y+z)$

$z = frac{1}{2}(y+z) -frac{1}{2}(x+y) + frac{1}{2}(z+x)$

So $x,y,z$ can be written as a linear combination of $(x+y),(z+x),(y+z)$ , hence the spans are equal.