Linear Algebra: Finding the Standard Matrix from a Function

[Cod]

Homework Statement

Find the standard matrix of T(f(t)) = f(3t-2) from P2 to P2.

Homework Equations

n/a

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.

 

[vela]

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, t²}, the first column of the matrix would correspond to T(1), the second column to T(t), and the third column to T(t²).

 

[Cod]

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.

 

[vela]

Your question doesn't make sense. Rⁿ is a vector space, not a basis.

 

[Cod]

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

 

[vela]

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

 

[Cod]

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

So, I could choose a basis of 1, t, t2; 1, t, t2, t3; and so on (to tnth)?

 

[vela]

Not for Rⁿ because those aren't vectors in Rⁿ. P2 consists of polynomials of degree less than or equal to 2, and each polynomial is a linear combination of 1, t, and t². It turns out 1, t, and t² are also independent, so they form a basis for P2. Rⁿ, however, is a set of n-tuples, not polynomials. You need a collection of n linearly independent n-tuples to have a basis for Rⁿ.