Proof, intersection and sum of vector spaces

In summary, to prove that V^{\bot}\cap W^{\bot}=(V+W)^{\bot}, we need to show that an element in the left-hand side is also in the right-hand side and vice versa. This can be done by understanding the definitions and discovering any tricky details. It is important to work through the steps to fully grasp the implications. We can start by understanding that a vector v in the subspace U^\perp is perpendicular to all elements of the subspace U. We can also use the fact that v^Tx=0 for x\in V and w^Ty=0 for y\in W.
  • #1
lukaszh
32
0
Hello,
how to prove this:
[tex]V^{\bot}\cap W^{\bot}=(V+W)^{\bot}[/tex]
Thanks
 
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  • #2
It is a "simple" matter of proving an element in the left-hand side is in the right-hand side and vis versa by parsing the definitions. But you'll learn little by seeing it done. You need to go through the steps of discovering the tricky details and resolving them so you appreciate the implications.
 
  • #3
Could you show me, how to do it?
 
  • #4
lukaszh said:
Could you show me, how to do it?

Yes but I'd rather you show me some start first. I take it this is an assignment in studying linear algebra. The point of an assignment if for you to puzzle through the problem and thereby learn.

I'll start you by pointing out that if a vector [itex]v[/itex] is in the subspace [itex]U^\perp[/itex] then it must be perpendicular to all elements of the subspace [itex]U[/itex].
 
  • #5
I know this:
[tex]\left(v\in V^{\bot}\wedge v\in W^{\bot}\right)\Rightarrow\left(v\in V^{\bot}\cap W^{\bot}\right)[/tex]
[tex]\left(x\in V\wedge x\in W\right)\Rightarrow\left(x\in V\cap W\right)[/tex]
I can also write that
[tex]v^Tx=0\,;\; x\in V, v\in V^{\bot}[/tex]
[tex]w^Ty=0\,;\; y\in W, w\in W^{\bot}[/tex]
 

What is a vector space?

A vector space is a mathematical structure that consists of a set of objects called vectors, and a set of operations that can be performed on these vectors. These operations include addition and scalar multiplication, which must obey certain axioms or rules. Vector spaces are important in many areas of mathematics and science, including linear algebra, geometry, and physics.

What is the proof of a vector space?

The proof of a vector space is a mathematical argument that shows that a given set and set of operations satisfy the axioms or rules of a vector space. This is typically done by showing that the operations of addition and scalar multiplication satisfy the axioms, such as closure, associativity, and distributivity. The proof is important because it ensures that the set and operations are valid and can be used to solve problems.

What is the intersection of vector spaces?

The intersection of two vector spaces, V and W, is the set of all elements that are in both V and W. In other words, it is the set of all vectors that are common to both spaces. The intersection of vector spaces is important because it allows us to find common solutions to problems in both spaces, and it also helps us to understand the relationship between different vector spaces.

What is the sum of vector spaces?

The sum of two vector spaces, V and W, is the set of all possible combinations of vectors from V and W. This includes both addition and scalar multiplication operations. The sum of vector spaces is important because it allows us to combine different vector spaces and create new spaces with different properties and dimensions.

How do you determine if two vector spaces are equal?

To determine if two vector spaces, V and W, are equal, we need to show that they have the same elements and the same operations. This means that every vector in V must also be in W, and vice versa, and that the operations of addition and scalar multiplication must produce the same results in both spaces. If these conditions are met, then the two vector spaces are considered equal.

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