Rank Of Gram Matrix

Let v 1 , , v n R m v_1, \ldots, v_n \in \mathbb{R}^m be a collection of vectors. TheGram matrixof this collection is defined to be the n n -by- n n matrix whose entry in the i th i^\text{th} row and j th j ^ \文本{th} column is a i j = v i v j a_{ij} = v_i \cdot v_j , where \text{}\cdot denotes thedot product.

Consider the Gram matrix G G of the collection:

v 1 = ( 1 , 2 , 1 ) v 2 = ( 3 , 5 , 1 ) v 3 = ( 0 , 3 , 6 ) v 4 = ( 4 , 2 , 0 ) . \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 ? G?

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