Direct Sum of Vectors in R^4: Determine Which Sums Are Direct and Equal to R^4

[Jamo1991]

Homework Statement

In R^4 which of the following sums U+V, U+W and V+W are direct? Give reasons
And which of these sums equal R^4?

Homework Equations

U = {(0, a, b, a-b) : a,b ∈ R}
V = {(x, y, z, w) : x=y, z=w}
W = {(x, y, z, w) : x=y}

The Attempt at a Solution

I put that none are direct sums as U is the only one to contain the zero vector meaning that none of the intersections would also be able to contain the zero vector. Is this right? It seems too simple.

For the second part I am unsure where to start.

 

[HallsofIvy]

Why do you say that V and W do not contain the 0 vector? V is the set of all (x, y, z, w) with x= y, z= w or, more simply with (x, x, z, z). Since x and z can be any numbers take x= z= 0. And why do mention "intersections"? This question is about direct sums, not intersections.

What is the definition of direct sum?

 

[HallsofIvy]

Yes, U+ V is a direct sum. That means that the dimension of U+V is the dimension of U plus the dimension of V. What are those?

Notice that any vector in U is of the form (0, a, b, a-b)= (0, a, 0, a)+ (0, 0, b, -b) and that any vector in V is of the form (a, a, b, b)= (a, a, 0, 0)+ (0, 0, b, b).

 

[HallsofIvy]

No, the dimensions are NOT 4. Do you really understand what "dimension" means? U, V, and W are all subspaces of $R^4$. Can you give a basis for each vector space?