Linear Dependence In R\mathbb{R} Over Q\mathbb{Q}

Algebra Level 4

Let VV be a vector space over any field FF.

A collection of vectors v1,v2,,vnVv_1, v_2, \ldots, v_n \in V is called dependent if there exist real numbers a1,a2,,anRa_1, a_2, \ldots, a_n \in \mathbb{R} such that a1v1++anvn=0a_1 v_1 + \cdots + a_n v_n = 0 and at least one of the aia_i's is nonzero. Consequently, a collection of vectors is called independent if it is not dependent.

Note that R\mathbb{R} may be thought of as a vector space over the field Q\mathbb{Q} of rational numbers. With this vector space structure on R\mathbb{R}, is the set {2,3,5}R\big\{\sqrt{2}, \sqrt{3}, \sqrt{5}\big\} \subset \mathbb{R} dependent or independent? What about the set {π4,arctan(32),arctan(2)}R?\left\{\frac{\pi}{4}, \arctan\left(\frac{3}{2} \right), \arctan(2) \right\} \subset \mathbb{R}?

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