How to show U and W direct sum of V?

[cookiesyum]

Let V = R^3. Let W be the space generated by w = (1, 0, 0), and let U be the subspace generated by u_1 = (1, 1, 0) and u_2 = (0, 1, 1). Show that V is the direct sum of W and U.

 

[Dick]

Try and follow the forum guidelines, ok? Show an attempt or state what's confusing you. Don't just post the question. You want to show each vector in V=R^3 can be expressed as the sum of a vector in U and a vector in W in a unique way. Try it.

 

[cookiesyum]

Try and follow the forum guidelines, ok? Show an attempt or state what's confusing you. Don't just post the question. You want to show each vector in V=R^3 can be expressed as the sum of a vector in U and a vector in W in a unique way. Try it.

Yes, I will. I'm sorry about the misunderstanding, I'm new to this forum.

First I let v = (2, 2, 1) E in R^3. Then I showed that v = u1 + u2 + w = (1 1 0) + (0 1 1) + (1 0 0) = (2 2 1) which still E in R^3. Now, how do I show that the intersection of U and W contains only 0. Do I do this by showing u1, u2, and w are linearly independent?

 

[Dick]

Well, you've only shown that one particular vector can be expressed as a sum, but you have show that way of decomposing v is unique. But, yes, if you show u1, u2 and w are linearly independent than you know that the coefficients are unique, and since there are 3 vectors, they span all of R^3.