# Help with part of my Linear Algebras project - affine sets and mappings

• TheRookie
In summary, the conversation discusses two questions related to affine mappings and their properties. The first question states that an affine mapping, f: V→W, can be expressed as f(x) = Tx + b, where T: V→ W is linear and b∈W, and it must satisfy the condition f(λx + (1-λ)y) = λf(x) + (1-λ)f(y) for all x and y in V, and λ in ℝ. The second question asks to prove that the image of an affine subset under an affine mapping is affine, and also that the composition of two affine maps is affine. The conversation also suggests understanding the informal meaning of an affine map to better understand its
TheRookie

## Homework Statement

I'm a 2nd year undergraduate student, so I suppose many users here won't find this too difficult, but I've had some issues with the following questions and, of course, any help would be very much appreciated:

(i) Prove f: V→W is affine (where V and W are real vector space) iff it is of the form f(x) = Tx + b, where T: V→ W is linear and b∈W. Prove T and b are uniquely determined by f.

(ii) Prove the image of an affine subset under an affine mapping is affine. Prove the composition of two affine maps is affine.

## The Attempt at a Solution

(i) If f: V→W is an affine mapping, we must have f(λx + (1-λ)y) = λf(x) + (1-λ)f(y) for all x and y in V, and λ in ℝ.
This is what immediately came to mind, but I can't see a connection between this and the question...

(ii) I'm sorry, but I've gone over this again and again and don't know where to start.Regards,
Pete

Try substituting the condition on 3i) into f(x) in 1i), and see if it is satisfied. I think it may help you over the

long run if you have a good feel for the meaning of an Affine map. Not that you should

drop rigor at all, but it helps if you understand what is going on at an informal level.

Basically, to start, an affine map is a translation/shift of a linear map.

This thread has been closed because of academic misconduct.

## 1. What are affine sets in Linear Algebra?

Affine sets in Linear Algebra are subsets of a vector space that are closed under linear combinations and translations. In other words, they are sets that maintain their shape and structure when translated or stretched in any direction.

## 2. How are affine sets different from vector spaces?

Affine sets and vector spaces are similar in that they both involve linear combinations of vectors. However, unlike vector spaces which contain the origin, affine sets do not necessarily contain the origin and are instead translated from some reference point.

## 3. What is an affine mapping?

An affine mapping is a function that preserves affine structure, meaning it maps affine sets to affine sets. It is composed of a linear transformation followed by a translation, and can be represented by a matrix-vector multiplication.

## 4. How do I determine if a mapping is affine?

To determine if a mapping is affine, you can check if it preserves affine structure by verifying that it maps affine combinations to affine combinations. Additionally, an affine mapping must map the origin to itself and preserve parallelism between lines and planes.

## 5. How are affine sets and mappings used in real-world applications?

Affine sets and mappings have numerous applications in fields such as computer graphics, economics, and optimization. They are used to model geometric transformations, linear programming problems, and economic equilibrium, among others.

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