Let $A = span\{a_1,a_2\}=\{x_1a_1+x_2a_2 | x_1,x_2 \in \mathbb{R}\}$ What are the elements of $H$ if $H$ is the quotient of A by the one dimensional subspace spanned by $a_1+ a_2$.
Elements of a Quotient set
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abstract-algebra
group-theory
vector-spaces
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0Can you provide any details about why you can't solve the problem yourself? – 2017-01-24
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0I know that the elements of the one dimensional subspace spanned looks like $\{xa_1+xa_2, x \in \mathbb{R}\}$. – 2017-01-24
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0I am just confused with the division. – 2017-01-24
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0What is your definition of a quotient space? – 2017-01-24
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0$H = A/B = \{a+B, a \in A\}$. – 2017-01-24
1 Answers
1
You actually have some freedom of how you represent element of $A/H$. As you are using additive notation I will assume all groups concerned are abelian (though up to notation there's not a lot of difference).
In any quotient $G/N$, as you noted in a comment, elements look like $g+N$ where $g\in G$. I suspect you want some way of uniquely representing elements of $H/A$.
Well, consider $x_1a_1+x_2a_2+H$ with $x_1,x_2\in\mathbb{R}$. Since $x_2(a_1+a_2)\in H$ we have $x_1a_1+x_2a_2+H=x_1a_1+x_2a_2-x_2(a_1+a_2)+H=(x_1-x_2)a_1+H$. That is, every element of $A/H$ can be represented by $xa_1+H$ for some $x\in\mathbb{R}$.
It's an easy exercise to show that this $x$ is unique and you can play the same game to represent elements of $A/H$ uniquely as $xa_2+H$ or in fact $x(x_1a_1+x_2a_2)+H$ for any choice of $x_1,x_2\in\mathbb{R}$ with $x_1\ne x_2$.