# What Is the Dimension of Sp{v1,..,vk,u,w}?

• nhrock3
In summary, there are k+1 vectors in space V, with the group {v1,..,vk,u} being independent. The dimension of sp{v1..vk,u,w} is equal to the sum of the dimensions of sp{v1..vk} and sp{u,w} minus the dimension of their intersection. Since u is independent of v1,v2,...,vk, the dimension of sp{v1..vk} is equal to k. However, the dimension of sp{u,w} is unknown.
nhrock3
there is
v1,..,vk,u,w vectors on space V
the group {v1,..,vk,u} is independant
and
$$Sp\{v1,..,vk\}\cap Sp\{u,w\}\neq\{0\}$$
find the dimention of sp{v1..vk,u,w}
?

if {v1,..,vk,u} is independant then its span dimention is k+1
and span so {v1,..,vk} independat to and dim sp{v1,..,vk}=k

the dimention of sp{v1..vk,u,w} is
dim(sp{v1..vk,u,w})=dim(sp{v1..vk}and sp{u,w}})=dim(sp{v1..vk}+sp{u,w})=dim(sp{v1..vk})+dim(sp{u,w})-dim(sp{v1..vk} intersect sp{u,w}})

if v1..vk,u is independant then v1,..,vk is independant too so dim (sp{v1..vk})=k

now regardingthe other two members i don't know

You are given that u is indendent of v1,v2,...,vk.
What does this imply about the intersection of the sapces spanned by {v1,v2...,vk} and {u}?

## 1. What is the definition of "dimension of group"?

The dimension of a group refers to the number of independent parameters needed to describe the group's elements. It is the minimum number of generators required to generate the group.

## 2. How is the dimension of a group determined?

The dimension of a group can be determined by finding the number of elements in the group and then using the group's defining properties to determine the minimum number of generators needed to generate all of the elements.

## 3. Can the dimension of a group change?

Yes, the dimension of a group can change if the group undergoes a transformation or if new elements are added to the group. However, for a given group, the dimension remains constant.

## 4. What is the relationship between the dimension of a group and its order?

The dimension of a group is not directly related to its order. The order of a group refers to the number of elements in the group, while the dimension refers to the number of parameters needed to describe those elements. However, the order can give some insight into the possible dimensions of a group.

## 5. Why is the dimension of a group important in mathematics?

The dimension of a group is important because it helps us understand the structure and properties of the group. It is also a fundamental concept in many branches of mathematics, such as group theory, algebra, and geometry. Additionally, the dimension of a group can be used to classify and compare different groups.

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