Prove sum of two subspaces is R^3

In summary, to prove that the sum of subspaces U and W is R^3, we need to show that any vector in R^3 can be written as the sum of a vector from U and a vector from W. This can be done by finding bases for U and W and combining them to form a basis for R^3. We can also use the dimension theorem and the fact that if two subspaces have the same dimension, they are equal.
  • #1
tjkubo
42
0
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?
 
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  • #2
tjkubo said:
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?
 
  • #3
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.
 
  • #4
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...)
 
  • #5
tjkubo said:
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?
 
  • #6
Moreover, notice that Sp(B1)+Sp(B2)=Sp(B1uB2), where B1,B2 are sets or bases (in our case).
 
  • #7
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.
 

What is the definition of "sum of two subspaces" in a vector space?

The sum of two subspaces in a vector space is the set of all possible combinations of vectors from each subspace. In other words, it is the subset of the vector space that is spanned by both subspaces.

How is the sum of two subspaces calculated?

To calculate the sum of two subspaces, you first need to find a basis for each subspace. Then, you can add the basis vectors from each subspace together to create a new set of vectors. This new set of vectors will be the basis for the sum of the two subspaces.

What is the dimension of the sum of two subspaces?

The dimension of the sum of two subspaces is the same as the sum of the dimensions of the individual subspaces, minus the dimension of their intersection. In the case of R^3, the dimension of the sum of two subspaces will be at most 3.

How do you prove that the sum of two subspaces is equal to R^3?

To prove that the sum of two subspaces is equal to R^3, you need to show that every vector in R^3 can be written as a linear combination of vectors from each subspace. This can be done by showing that the basis for the sum of the two subspaces spans the entire vector space R^3.

What are some real-world applications of the concept of sum of two subspaces in R^3?

The concept of sum of two subspaces in R^3 can be applied in various fields such as engineering, physics, and economics. For example, in engineering, this concept can be used to analyze the forces acting on a structure by breaking them down into different subspaces and then finding the sum of those subspaces. In physics, this concept can be used to understand the motion of multiple objects by breaking them down into different subspaces and then finding the sum of those subspaces. In economics, this concept can be used to analyze the relationship between different variables by breaking them down into different subspaces and then finding the sum of those subspaces.

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