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Linear Algebra: Finding the Standard Matrix from a Function 
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#1
Nov610, 01:33 PM

P: 308

1. The problem statement, all variables and given/known data
Find the standard matrix of T(f(t)) = f(3t2) from P2 to P2. 2. Relevant equations n/a 3. The attempt at a solution The overall question has to do with finding the determinants, so the matrix is provided; however, I want to know how the author came up with the standard matrix of T. Any help is greatly appreciated. 


#2
Nov610, 01:53 PM

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PF Gold
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You can find the columns of the matrix representation by applying the transformation to the basis vectors. If you're using the basis {1, t, t^{2}}, the first column of the matrix would correspond to T(1), the second column to T(t), and the third column to T(t^{2}).



#3
Nov710, 04:33 PM

P: 308

Thanks for the guidance. What happens when the basis is Rn? I realize R2 is a 3x3 matrix, R3 is a 4x4, and so on.



#4
Nov710, 04:36 PM

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Linear Algebra: Finding the Standard Matrix from a Function
Your question doesn't make sense. R^{n} is a vector space, not a basis.



#5
Nov710, 07:05 PM

P: 308

Sorry, I was looking at my homework when I typed the last post. I meant vector space.



#6
Nov710, 08:05 PM

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Same thing. You choose a basis and apply the transformations to the basis vectors to get the columns of the matrix representing the transformation.



#7
Nov810, 04:50 PM

P: 308




#8
Nov810, 05:13 PM

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Not for R^{n} because those aren't vectors in R^{n}. P2 consists of polynomials of degree less than or equal to 2, and each polynomial is a linear combination of 1, t, and t^{2}. It turns out 1, t, and t^{2} are also independent, so they form a basis for P2. R^{n}, however, is a set of ntuples, not polynomials. You need a collection of n linearly independent ntuples to have a basis for R^{n}.



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