Concept of a basis for a vector space

In summary, a "basis" for a vector space refers to a set of vectors that can be used to express any vector within that space through linear combinations. In the example provided, the basis for subspaces U and V are represented by u1 and v1 respectively, and their intersection is shown to be a linearly independent set in the larger vector space W.
  • #1
mrroboto
35
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concept of a "basis" for a vector space

I do not understand the concept of a "basis" for a vector space.

Here's an example from my practice final exam:

Suppose U and V are subspaces of the real vector space W and {u1} is a basis for U and {v1} is a basis for V. If U intersection V = {0} show that {u1, v1} is a linearly independent set it W.

I probably need additional help with this example, but if someone could explain a "basis" to me in terms of this example I would greatly appreciate it.

Thanks.
 
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  • #2
Any vector in U can be expressed as a linear transformation of u1 (e.g., k*u1). Any vector in V can be expressed as a linear transformation of v1 (e.g., k*v1).
 
  • #3
Ok, thanks!
 

FAQ: Concept of a basis for a vector space

What is a vector space?

A vector space is a mathematical structure that consists of a set of objects, called vectors, that can be added together and multiplied by numbers, called scalars.

What is a basis for a vector space?

A basis for a vector space is a set of vectors that are linearly independent and span the entire space. This means that any vector in the space can be written as a unique combination of the basis vectors.

Why is a basis important in a vector space?

A basis is important because it provides a way to represent any vector in the space using a finite set of vectors. This makes it easier to perform calculations and transformations on vectors in the space.

How do you determine if a set of vectors is a basis for a vector space?

To determine if a set of vectors is a basis for a vector space, you can check if the vectors are linearly independent and if they span the entire space. You can also use the dimension of the space to determine the number of linearly independent vectors needed for a basis.

Can a vector space have more than one basis?

Yes, a vector space can have multiple bases. In fact, any set of linearly independent vectors in the space can be considered a basis. However, all bases for a given vector space will have the same number of vectors, which is called the dimension of the space.

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