# Define the set C with given operations (Linear Algebra)

Define the set $$\{C = (x,y) : x,y \in \Bbb R\}$$ with the operations as $$(x_1, y_1) \oplus (x_2, y_2) = (x_1 + x_2 + 1, y_1 + y_2 + 1)$$ and $$\alpha$$ $$\otimes$$ ($$x_1$$, $$y_1$$) = ($$\alpha$$ $$x_1$$ + $$\alpha$$ - 1, $$\alpha$$ $$y_1$$ + $$\alpha$$ - 1), for all ($$x_1$$, $$y_1$$), ($$x_2$$, $$y_2$$) $$\epsilon$$ C and $$\alpha$$ $$\epsilon$$ $${R}$$: Is it a vector space? Justify your answer.

I understand how to figure out if something is a vector space, I'm mostly confused by the use of the direct addition and multiplication signs because I was under the impression that the direct addition of ($$x_1$$, $$y_1$$) $$\oplus$$ ($$x_2$$, $$y_2$$) is always ($$x_1$$ + $$x_2$$, $$y_1$$ + $$y_2$$). So how would I define the set in this situation?

• The notation $\oplus$ in the question does not correspond to your notion of direct sum. It is just a notation for anew operation. – Kavi Rama Murthy Feb 4 '20 at 5:25
• Just as $\otimes$ does not have its usual meaning as a tensor product in this context, so does $\oplus$ not have its usual meaning as a direct sum. – Ben Grossmann Feb 4 '20 at 7:46

Hint: (For an expedient proof). Note that for $$x_1,x_2 \in \Bbb R$$, $$\alpha x_1 + \alpha - 1 = \alpha(x_1 + 1) - 1\\ x_1 + x_2 + 1 = (x_1 + 1) + (x_2 + 1) - 1.$$
Another perspective: if $$\phi:\Bbb R \to \Bbb R$$ is defined by $$\phi(x) = x - 1$$, then we have $$(x_1,x_2) \oplus (y_1,y_2) = (f^{-1}(f(x_1) + f(x_2)), f^{-1}(f(y_1) + f(y_2))),\\ \alpha \otimes (x_1 ,x_2) = (f^{-1}(\alpha f(x_1)), f^{-1}(\alpha(x_2))).$$ In other words, $$\oplus$$ and $$\otimes$$ are addition and mulitplication redefined via a transport of structure. See this question and this question for the same idea with a more complicated $$f:\Bbb R \to \Bbb R$$.