# [LinAlg] Show that T:C[a,b] -> R is a linear transformation

• bornofflame
In summary, the problem asks to show that the mapping T:C[a,b] -> is a linear transformation, which means that it follows the rules of i. T(u+v) = T(u) + T(v) for all u, v in V, and ii. T(cu) = cT(u) for all u in V and all scalars c. The attempt at a solution shows that the mapping T(f) = ∫(from a to b) f(x)dx follows these rules, making it a linear transformation. The use of Latex can improve the presentation of mathematical expressions.
bornofflame

## Homework Statement

Show that T:C[a,b] -> defined by T(f) = ∫(from a to b) f(x)dx is a linear transformation.

## Homework Equations

Definition of a linear transformation:
A linear transformation T from a vector space V into a vector space W is a rule that assigns to each vector x in V a unique vector T(x) in W, such that
i. T(u+v) = T(u) + T(v) for all u, v in V, and
ii. T(cu) = cT(u) for all u in V and all scalars c

## The Attempt at a Solution

In order for this mapping to be a linear transformation it must follow the above rules, so
i. T(f + g) = ∫ [f(x) + g(x)]dx = ∫ f(x)dx + ∫ g(x)dx = T(f) + T(g)
ii. T(cf) = ∫ cf(x)dx = c∫ f(x)dx = cT(x)
Is it really that simple or did I over simplify the problem?

This problem is not taken from any book (afaik) and doesn't resemble much that we've done in class or from the book so it's throwing me off a bit as I bend my brain to look at things from different angles trying to make sure that I don't miss anything.

bornofflame
PeroK said:
Yes, it really is as simple as that.

You should try to learn a bit of Latex:

https://www.physicsforums.com/help/latexhelp/

For example:

##\int_a^b f(x)dx##

Looks a lot better!

It does! I'll look into that. Thanks!

## 1. What is a linear transformation?

A linear transformation is a mathematical function that maps a vector space to another vector space while preserving the properties of vector addition and scalar multiplication. In other words, it takes in a vector and produces another vector that is a linear combination of the input vector.

## 2. How do you show that T:C[a,b] -> R is a linear transformation?

To show that T:C[a,b] -> R is a linear transformation, we must prove that it satisfies the two properties of linearity: additivity and homogeneity. Additivity means that T(u+v) = T(u) + T(v), where u and v are vectors in the vector space C[a,b]. Homogeneity means that T(cu) = cT(u), where c is a scalar and u is a vector in C[a,b]. If T satisfies these two properties, then it is a linear transformation.

## 3. What is the significance of T:C[a,b] -> R in linear algebra?

T:C[a,b] -> R is a linear transformation that maps a continuous function in the interval [a,b] to a real number. It is significant in linear algebra because it helps us understand how linear transformations work in the context of functions. It also allows us to apply linear algebra concepts and techniques to solve problems involving functions.

## 4. What are some examples of linear transformations?

Some examples of linear transformations include translation, rotation, scaling, and shearing. In linear algebra, we often deal with transformations that involve matrices, such as rotation matrices, scaling matrices, and projection matrices. These transformations are considered linear because they satisfy the properties of linearity.

## 5. How can we use linear transformations in real-world applications?

Linear transformations have many real-world applications, such as image and signal processing, computer graphics, data compression, and machine learning. For example, in image processing, linear transformations are used to rotate, scale, and enhance images. In machine learning, linear transformations are used to preprocess data and extract features for training models.

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