Linear Algebra: Finding the Standard Matrix from a Function

In summary, the conversation discusses finding the standard matrix of a transformation from P2 to P2, with the advice to find the columns of the matrix by applying the transformation to the basis vectors. It is noted that Rn is not a basis, and the basis for P2 is 1, t, and t2. The conversation also clarifies that Rn is a set of n-tuples, not a basis.
  • #1
Cod
325
4

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.
 
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  • #2
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).
 
  • #3
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
Your question doesn't make sense. Rn is a vector space, not a basis.
 
  • #5
Sorry, I was looking at my homework when I typed the last post. I meant vector space.
 
  • #6
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
vela said:
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)?
 
  • #8
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.
 

1. What is linear algebra?

Linear algebra is a branch of mathematics that deals with the study of linear equations and their representations in vector spaces. It involves the use of matrices and vectors to solve problems involving systems of linear equations.

2. What is a standard matrix?

A standard matrix is a matrix that represents a linear transformation from one vector space to another. It is a square matrix that has the same number of rows and columns and contains the coefficients of the linear transformation.

3. How do you find the standard matrix from a function?

To find the standard matrix from a function, you need to first identify the input and output variables of the function. Then, construct a matrix with the coefficients of the input variables as the columns and the coefficients of the output variables as the rows. This matrix is the standard matrix for the given function.

4. What is the importance of finding the standard matrix?

Finding the standard matrix allows us to easily perform operations such as composition, inverse, and multiplication on linear transformations. It also helps us to better understand the properties of linear transformations and their effects on vector spaces.

5. Are there any applications of finding the standard matrix?

Yes, there are many applications of finding the standard matrix. It is used in computer graphics, robotics, image processing, and machine learning, among others. It is also essential in solving systems of linear equations and analyzing the behavior of linear systems in engineering and physics.

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