- #1
lukaszh
- 32
- 0
Hello,
how to prove this:
[tex]V^{\bot}\cap W^{\bot}=(V+W)^{\bot}[/tex]
Thanks
how to prove this:
[tex]V^{\bot}\cap W^{\bot}=(V+W)^{\bot}[/tex]
Thanks
lukaszh said:Could you show me, how to do it?
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.
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.
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.
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.
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.