Showing that a non-empty subset M is an affine subset of R^4

**juanma101285** · March 11th 2012, 07:14 AM

Hi, I have to do a project on affine subsets and affine mappings, but I have no clue what they are... We are given only one clue and I can't find many notes on google. I would really appreciate it if someone could help me with this first problem (and if you could also give me a link to some good notes on the topic). Thanks a lot!

The problem says:

"To illustrate this concept, show that

$M=\{x=(x_1, ..., x_4) \epsilon \mathbb{R}^4 : 2x_1-x_2+x_3=1$ and $x_1+4x_3-2x_4=3\}$

is an affine subset of $\mathbb{R}^4$ ."

And the only hint we get is

"Throughout Part A of this project, V will be a real vector space and, for a non-empty subset S of V and a $\epsilon$ V, the set { $x+a : x \epsilon S$ } will be denoted by S+a.

An affine subset of V is a non-empty subset M of V with the property that $\lambda *x+(1- \lambda )*y \epsilon M$ whenever $x,y \epsilon M$ and $\lambda \epsilon \mathbb{R}$ ."

**juanma101285** · March 11th 2012, 08:08 AM

OK, I have managed to solve the problem... I just have to substitute lambda*x+(1-lambda)*y into both of the conditions given.

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