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an equivalence relation in a vector subspace of R^3

Source: Spanish Mathematical Olympiad 1972 P8

December 5, 2022
linear algebraalgebraequivalence relation

Problem Statement

We know that R3={(x1,x2,x3)∣xi∈R,i=1,2,3}R^3 = \{(x_1, x_2, x_3) | x_i \in R, i = 1, 2, 3\} is a vector space regarding the laws of composition (x1,x2,x3)+(y1,y2,y3)=(x1+y1,x2+y2,x3+y3)(x_1, x_2, x_3) + (y_1, y_2, y_3) = (x_1 + y_1, x_2 + y_2, x_3 + y_3), λ(x1,x2,x3)=(λx1,λx2,λx3)\lambda (x_1, x_2, x_3) = (\lambda x_1, \lambda x_2, \lambda x_3), λ∈R\lambda \in R. We consider the following subset of R3R^3 : L={(x1,x2,x3)∈R3∣x1+x2+x3=0}L =\{(x_1, x2, x_3) \in R^3 | x_1 + x_2 + x_3 = 0\}. a) Prove that LL is a vector subspace of R3R^3 . b) In R3R^3 the following relation is defined x‾Ry‾⇔x‾−y‾∈L,x‾,y‾∈R3\overline{x} R \overline{y} \Leftrightarrow \overline{x} -\overline{y} \in L, \overline{x} , \overline{y} \in R^3. Prove that it is an equivalence relation. c) Find two vectors of R3R^3 that belong to the same class as the vector (−1,3,2)(-1, 3, 2).
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