Prove sum of two subspaces is R^3

[tjkubo]

How do you prove that the sum of the following subspaces is R^3?
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?

 

[Hurkyl]

know intuitively that U+W=R^3 because U is a plane and W is a line not contained in U

How does that work?

 

[tjkubo]

x-y-z=0 is the equation of a plane and n=(1,-1,-1) is a normal vector, so (1,-1,-1)t is a line perpendicular to the plane, and the span of the basis elements is R^3.

 

[Hurkyl]

That sounds like a sketch for a mathematical proof to me. Can you justify each part of it?

(p.s. nobody said anything about a basis...)

 

[HallsofIvy]

How do you prove that the sum of the following subspaces is R^3?
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?

Any vector in U can be written as <x, y, x- y>= <x, 0, x>+ <0, y, -y>= x<1, 0, 1>+ y<0, 1, -1>. What is a basis for U?

Any vector in W can be written as <t, -t, -t>= t<1, -1, -1>. What is a basis for U?

Suppose you put those two bases together? Are they still independent?

 

[boaz]

Moreover, notice that Sp(B1)+Sp(B2)=Sp(B1uB2), where B1,B2 are sets or bases (in our case).

 

[elibj123]

You can also use the dimension theorem dim(U+V)=dim(U)+dim(V)-dim(U^V)

And the fact that if U is a subspace of V and they have the same dimension then U=V.