Standard representation of a vector space

[quasar_4]

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! Any enlightenment would be wonderful.

Thanks.

 

[AKG]

Given a vector v and an ordered basis {e₁, ..., e_n}, there are unique field elements f¹, ..., fⁿ such that v = f¹e₁ + ... + fⁿe_n. 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. $\beta$. Repeat this exercise, this time letting e₁ = (1,2) and e₂ = (3,4). Find the field elements, etc...

 

[tacman]

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.