Register to reply 
Linear Algebra: Finding the Standard Matrix from a Function 
Share this thread: 
#1
Nov610, 01:33 PM

P: 307

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

Emeritus
Sci Advisor
HW Helper
Thanks
PF Gold
P: 11,687

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: 307

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

Emeritus
Sci Advisor
HW Helper
Thanks
PF Gold
P: 11,687

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: 307

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



#6
Nov710, 08:05 PM

Emeritus
Sci Advisor
HW Helper
Thanks
PF Gold
P: 11,687

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: 307




#8
Nov810, 05:13 PM

Emeritus
Sci Advisor
HW Helper
Thanks
PF Gold
P: 11,687

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}.



Register to reply 
Related Discussions  
Linear Algebra Commuting matrix  Calculus & Beyond Homework  2  
Linear Algebra Diagonal Matrix  Calculus & Beyond Homework  2  
Linear algebra: Finding a linear system with a subspace as solution set  Calculus & Beyond Homework  1  
Elementary Linear Algebra (matrix)  Calculus & Beyond Homework  1  
Matrix, making R2 to R3, finding standard matrix A! where did i mess up?  Calculus & Beyond Homework  1 