# Integration as a Linear Transformation

• TranscendArcu
In summary, the problem asks for a linear transformation from the real numbers to the real numbers such that the function F(x) = 0 is satisfied.
TranscendArcu

## The Attempt at a Solution

I: P(R) → P(R) such that I(a0+a1x + ... + anxn) = 0 + a0x + (a1x2)/2 + ... + (an xn+1)/(n+1)

Clearly this is just integration such that c = 0. It is easily shown that integration is a linear transformation, so I conclude that I is a linear transformation.

However, in calculus we do not usually require that c = 0. Yet integration, if it is a linear transformation, must map the zero vector of the domain to the zero vector of the codomain. Let f(x) = a0 + a1x. Then F(x) = a0x + (a1x2)/2 + c. A polynomial is zero iff all of its coefficients are zero. If we decide that c is not necessarily zero, doesn't this iff fail? That is, is integration a linear transformation only when we stipulate that c=0?

Let U,V be vector spaces. A function T:U→V is said to be linear if T(ax+by)=aTx+bTy for all scalars a,b and all vectors x,y.

If U=V=ℝ, then the T defined by T(x)=ax+b is clearly not linear, unless b=0. (When U=V=ℝ, both the "scalars" and the "vectors" are members of ℝ). You're working with a different space, and I didn't even read your problem carefully, but my point is just that you need to use the definition of "linear" above, instead of your intuition about what a "line" is. That's why I'm mentioning that the only functions from ℝ to ℝ that are linear are the ones whose graphs are straight lines through the origin.

And doesn't integration satisfy your definition of linear?

∫(f(x) + g(x)) dx = ∫f(x)dx + ∫g(x)dx, for two functions, f,g, in P(R)
∫r(f(x))dx = r∫f(x)dx, for some r in R

Is that not true? And isn't it true that linear transformations must map the zero vector of the domain to the zero vector of the codomain? Maybe I'm misunderstanding.

Note that the statement of the problem specifically said "Let F(x) be the polynomial with F(0)= 0 such that F'= f".

If that condition "with F(0)= 0" were not there, this "theorem" would not be true.

Does that indicate that integration without the condition "F(0) = 0" is not a linear transformation?

TranscendArcu said:
Does that indicate that integration without the condition "F(0) = 0" is not a linear transformation?
Perhaps you can tell us. If f,g are arbitrary polynomials, a,b are arbitrary real numbers, and I denotes the map ##f\mapsto F##, what is I(af+bg)?

## What is integration as a linear transformation?

Integration as a linear transformation is a mathematical concept that involves finding the integral of a function, which is a mathematical operation that represents the area under a curve. This operation can be thought of as a linear transformation because it takes a function as an input and produces a new function as an output.

## How does integration relate to linear transformations?

Linear transformations are mathematical operations that preserve the basic shape and structure of a function. Integration, as a linear transformation, preserves the basic shape of a function while also finding the area under the curve. This means that the integral of a function can be thought of as a linear transformation of that function.

## What is the difference between integration and differentiation?

Integration and differentiation are inverse operations. While integration involves finding the area under a curve, differentiation involves finding the slope of a curve. In other words, integration is the inverse of differentiation and vice versa.

## How is integration used in real-world applications?

Integration has numerous real-world applications, especially in physics and engineering. For example, it can be used to calculate the work done by a force, find the center of mass of an object, or determine the volume of a three-dimensional shape. Additionally, integration is used in statistics to find the area under a probability density function, which is crucial for calculating probabilities and making predictions.

## What are some common techniques for solving integration problems?

Some common techniques for solving integration problems include using basic integration rules, such as the power rule and the substitution rule, as well as more advanced techniques like integration by parts, partial fractions, and trigonometric substitution. It is also helpful to have a good understanding of basic algebra and geometry when solving integration problems.

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