Rank Of Gram Matrix

Algebra Level 4

Let $v_1, \ldots, v_n \in \mathbb{R}^m$ 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 $\text{}\cdot$ denotes the dot product.

Consider the Gram matrix $G$ of the collection:

$\begin{aligned} v_1 &= (1,2,1)\\ v_2 &= (-3,5,1)\\ v_3 &= (0,-3,6)\\ v_4 &= (4,-2,0). \end{aligned}$

What is the rank of $G?$

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