Standard representation of a vector space

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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.
 

Answers and Replies

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AKG
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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...
 

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