Is Definite Integration from 0 to 1 a Linear Transformation from Pn to R?

For definite integration of functions in Pn, the linear transformation is between Rn+1 and R, where Rn+1 is the vector space of all polynomials of degree n or less and R is the vector space of real numbers. The transformation T is defined as T(a0, a1,...,an) = a0 + a1x + ... + anxn, which represents the definite integration from x=0 to x=1. This transformation is linear because it satisfies the properties T(kx) = kT(x) and T(x+y) = T(x) + T(y). The missing piece is that the transformation is not
  • #1
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I'm hoping I can get some help with the following question:

Does definite integration (from x = 0 to x = 1) of functions in Pn correspond to some linear transformation from Rn+1 to R?

OK, well my original answer was yes, but the textbook says "no, except for P0" which I do not understand.

So I have p(x) = anxn + an-1xn-1 + ... + a1x + a0

If I integrate from 0 to 1, I get: P(1) = an/(n+1) + an-1/n + ... + a1/2 + a0

Right?

So I have T(an, an-1, ..., a1, a0) = (an/(n+1) + an-1/n + ... + a1/2 + a0)

and T: Rn+1 -> R

And if I want to show that it is linear, then I show that the transformation has these properties:
T(kx) = kT(x) and
T(x+y) = T(x) + T(y)
and I think both cases are obvious.

And T is multiplication by [1/(n+1) | 1/n . . . 1/2 | 1]

What am I missing?
Thanks in advance for any help.
 
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  • #2
Linear transformations usually mean between vector spaces of the same dimension and therefore can be loosely thought of as matrices. what you have is a linear functional.

Mat
 

FAQ: Is Definite Integration from 0 to 1 a Linear Transformation from Pn to R?

What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space to another, preserving the structure of the original space. It is represented by a matrix and can be thought of as a transformation of points in space.

What are the basic properties of linear transformations?

The basic properties of linear transformations include preserving addition (T(u+v) = T(u) + T(v)), preserving scalar multiplication (T(cu) = cT(u)), and preserving the zero vector (T(0) = 0), among others.

How are linear transformations used in real-world applications?

Linear transformations are used in many fields, including physics, engineering, computer graphics, and data analysis. They can be used to model physical systems, manipulate and analyze data, and create visual representations of data.

What are the differences between linear and non-linear transformations?

A linear transformation preserves the structure of a vector space, while a non-linear transformation does not. Linear transformations can be represented by matrices, while non-linear transformations often cannot. Non-linear transformations can also exhibit more complex behavior, such as curvature or non-uniform scaling.

How can I determine if a transformation is linear?

To determine if a transformation is linear, you can check if it satisfies the basic properties of linearity. This includes checking if it preserves addition, scalar multiplication, and the zero vector. You can also represent the transformation as a matrix and check if it follows the rules of matrix multiplication.

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