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U = {(x,y,z) : x - y = z}

W = {(t,-t,-t) : t∈R}

I guess I need to show that any vector (x,y,z)∈R^3 can be written as the sum of a vector from U and a vector from W, but I'm not sure how to do that. I know intuitively that U+W=R^3 because U is a plane and W is a line not contained in U, but I don't know how to show that mathematically.

Any hints?