What is the quotient space $\mathbb{R}^2/F$ where $F =a[1,1]$?

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What is the quotient space $\mathbb{R}^2/F$ where $F =a[1,1]$? How do you find a basis of it? What is a good way to think of it in mind?

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If you meant the quotient space $\;U:=\Bbb R^2/F\;$ , then as $\;\dim U=1\;$ , you only need one non-zero vector to span it, and since

$$\binom10\notin F\implies \left\{\binom10+F\right\}\;\;\text{is a base for the quotient space}$$