Linear Transformation P2 > R^2

In summary, the question is asking to find a matrix A that maps from the space of polynomials of degree 2 to the space of 2-dimensional vectors, with the first coordinate being the integral of the polynomial over the interval [0,1] and the second coordinate being the value of the polynomial at 0. The solution involves finding the representation of L(a+bx) and using it to construct the desired matrix A.
  • #1
aredian
15
0

Homework Statement



If L( p(x) ) = [ integral (p(x)) dx , p(0) ]

find representation matrix A such that

L (a + bx) = A[a b]^T

Homework Equations





The Attempt at a Solution


I don't quite understand the question.
I think that:
if the base from p2 is {1, x} then any vector in p2 is of the form a + bx.
Then I can find L(a) = (a1 , a2) and L(bx) = (a3 , a4)
And use it to get A?

Can someone re phrase the question for me?
 
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  • #2
Given: [tex]L(p(x)) = \left[\begin{array}{c}\int p(x)dx\\p(0)\end{array}\right][/tex]

Put [tex]L(ax+b) = \left[\begin{array}{c}\int (ax+b)dx\\b\end{array}\right][/tex]

keeping going... the goal is to come up with a matrix A such that [tex]A\left[\begin{array}{cc}a&b\end{array}\right]^T[/tex] equals the right-hand side of the last equation.
 
  • #3
benorin said:
Given: [tex]L(p(x)) = \left[\begin{array}{c}\int p(x)dx\\p(0)\end{array}\right][/tex]

Put [tex]L(ax+b) = \left[\begin{array}{c}\int (ax+b)dx\\b\end{array}\right][/tex]

keeping going... the goal is to come up with a matrix A such that [tex]A\left[\begin{array}{cc}a&b\end{array}\right]^T[/tex] equals the right-hand side of the last equation.

[tex]L(ax+b) => \left[\begin{array}{cc}x&x^2 / 2 \\0&1\end{array}\right] [/tex] [tex][\begin{array}{c}\ \alpha\\ \beta \end{array} \right] [/tex]
 
  • #4
I think you're missing something important: the map is supposed to go from P^2 to R^2, so you can't map ax+b as you claim as the integral in the first coordinate yields (infinitely many) polynomial expressions. Surely the integral should be over some interval (I'd suggest the integral from 0 to 1 as the most likely).

Your matrix in the last post can't make sense, since it implies that the polynomial variables are allowed to appear in the positions in vectors in R^2.
 
  • #5
matt grime said:
I think you're missing something important: the map is supposed to go from P^2 to R^2, so you can't map ax+b as you claim as the integral in the first coordinate yields (infinitely many) polynomial expressions. Surely the integral should be over some interval (I'd suggest the integral from 0 to 1 as the most likely).

Your matrix in the last post can't make sense, since it implies that the polynomial variables are allowed to appear in the positions in vectors in R^2.

yup that was it.. I missed the interval, so the xs turned out to be 1's

thanks
 

1. What is a linear transformation from P2 to R^2?

A linear transformation from P2 to R^2 is a function that maps a polynomial of degree 2 to a point in two-dimensional space. It takes in a polynomial of the form ax^2 + bx + c and outputs a point (x, y) in the Cartesian plane.

2. How is a linear transformation represented mathematically?

A linear transformation from P2 to R^2 is represented by a 2x3 matrix [a b c] that maps the coefficients of the polynomial to the coordinates of the resulting point (x, y).

3. What are the properties of a linear transformation?

The properties of a linear transformation include:

  • Preserving addition: T(u + v) = T(u) + T(v)
  • Preserving scalar multiplication: T(cu) = cT(u)
  • Preserving zero vector: T(0) = 0

4. How does a linear transformation affect the shape of a polynomial?

A linear transformation from P2 to R^2 does not affect the shape of the polynomial, but rather maps it to a point in two-dimensional space. However, different transformations can result in different points, which can be used to represent the same polynomial in different ways.

5. How is a linear transformation used in real-world applications?

Linear transformations are used in a variety of real-world applications, including computer graphics, data compression, and data analysis. They can also be used to model physical systems, such as in physics and engineering. Additionally, linear transformations are used in machine learning algorithms, such as principal component analysis and linear regression, to reduce the dimensionality of data and make predictions.

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