# Rank Of Gram Matrix

Algebra Level 5

Let $$v_1, \cdots, v_n \in \mathbb{R}^m$$ be a collection of vectors. The Gram matrix of this collection is defined to be the $$n$$-by-$$n$$ matrix whose entry in the $$i^\text{th}$$ row and $$j^\text{th}$$ column is $$a_{ij} = v_i \cdot v_j$$, where $$\cdot$$ denotes the dot product.

Consider the Gram matrix $$G$$ of the collection: $v_1 = (1,2,1),$ $v_2 = (-3,5,1),$ $v_3 = (0,-3,6),$ $v_4 = (4,-2,0).$ What is the rank of $$G$$?

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