Standard representation of a vector space

In summary, the standard representation of a vector v with respect to an ordered basis is the unique set of field elements that can be multiplied by each basis vector to equal v. In the given example, the standard representations of the vector (1,-2) with respect to the ordered bases beta and gamma are (1,-2) and (-5,2), respectively. This is found by finding the field elements f1 and f2 that represent v in terms of the basis vectors and writing them in a tuple.
  • #1
quasar_4
290
0
Hi everyone,

Can anyone explain the following to me?

Given a basis beta for an n-dimensional vector space V over the field F, "the standard representation of V with respect to beta is the function phi_beta(x)=[x]_beta for each x in V." This is from my textbook.

It then proceeds to give the following example:

Let beta = {(1,0),(0,1)} and gamma = {(1,2),(3,4)}, where beta and gamma are ordered bases for R^2. For x=(1,-2), we have

phi_beta(x)=[x]_beta = (1,-2) and phi_gamma(x)=[x]_gamma = (-5,2).

I kind of see where the definition is going, and I understand how to find matrix representations of a transformation, but I just don't see what this standard representation thing is.

Where did the (1,-2) and the (-5,2) come from? How did they get these from the bases beta and gamma? I'm so confused! :confused: Any enlightenment would be wonderful.

Thanks.
 
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  • #2
Given a vector v and an ordered basis {e1, ..., en}, there are unique field elements f1, ..., fn such that v = f1e1 + ... + fnen. The standard representation, then, of v with respect to this ordered basis is (f1, ..., fn).

Take v = (1,-2), e1 = (1,0) and e2 = (0,1), then find the field elements f1, f2 such that v = f1e1 + f2e2. Write out (f1, f2), and this gives you the standard representation of v w.r.t. [itex]\beta[/itex]. Repeat this exercise, this time letting e1 = (1,2) and e2 = (3,4). Find the field elements, etc...
 
  • #3


The standard representation of a vector space is a way to represent vectors in that space using a specific basis. In this case, the basis is beta, which consists of the vectors (1,0) and (0,1). This means that any vector in the vector space V can be written as a linear combination of these two basis vectors. For example, the vector (1,-2) can be written as (1,0) - 2(0,1). This is where the (1,-2) came from in the example.

Similarly, the basis gamma consists of the vectors (1,2) and (3,4). So the vector (1,-2) can also be written as (-5/2)(1,2) + (3/2)(3,4), which is where the (-5,2) came from in the example.

The function phi_beta(x)=[x]_beta simply takes a vector x and expresses it in terms of the basis beta. So for the vector (1,-2), its standard representation with respect to beta is (1,-2). Similarly, for the vector (1,-2), its standard representation with respect to gamma is (-5,2).

I hope this helps clarify the concept of standard representation in vector spaces. Let me know if you have any further questions.
 

1. 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 scalars (such as numbers) to create new vectors. It is a fundamental concept in linear algebra and is used in many areas of mathematics and physics.

2. What is a standard representation of a vector space?

A standard representation of a vector space refers to a specific choice of basis vectors that can be used to represent all vectors in that space. This basis is usually chosen to be orthogonal (perpendicular) and normalized (unit length), which simplifies calculations and makes it easier to visualize and understand vectors in the space.

3. How do you determine the dimensions of a vector space?

The dimension of a vector space is equal to the number of basis vectors in its standard representation. For example, a two-dimensional vector space would have two basis vectors, while a three-dimensional vector space would have three basis vectors. In general, the dimension of a vector space is equal to the number of linearly independent vectors in the space.

4. Can the standard representation of a vector space be changed?

Yes, the standard representation of a vector space can be changed by choosing a different set of basis vectors. However, the underlying space itself does not change, only the way we represent vectors in that space. The choice of basis can affect calculations and interpretations, but the fundamental properties of the vector space remain the same.

5. How is a vector represented in a standard representation?

A vector in a standard representation is represented as a linear combination of the basis vectors. For example, in a two-dimensional vector space with basis vectors (1,0) and (0,1), the vector (2,3) can be represented as 2(1,0) + 3(0,1). This means that the vector has a component of 2 in the direction of the first basis vector and a component of 3 in the direction of the second basis vector.

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