Hello, hope you can help me to answer a question relatively short from one of my exercises about Vector spaces, I thank you in advance.

So I have not managed to show simply that set E is a linear subspace of

$\displaystyle \mathbb{R}^4.$, to give a basis and the dimension :

That is the statement:

Let $\displaystyle \{E=\{(a,b,b,a+b) \in \mathbb{R}^4;(a,b) \in \mathbb{R}^2\}.$

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What I 'd say :

Applying the method to show that a set is a linear subspace :

$\displaystyle E\subset \mathbb{R}^4$

$\displaystyle E \neq \emptyset$

To prove the stability of E for the addition and multiplication by a scalar, I considere 2 elements : "a" and "b" in E:

$\displaystyle \forall \lambda,\mu \in \mathbb{K}$ all that is necessary is to show that $\displaystyle \lambda.a+\mu.b \in E$

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In particular, I want you to help me to prove the stability of E.