Rank Of Gram Matrix

Algebra Level 4

Let v1,,vnRmv_1, \ldots, v_n \in \mathbb{R}^m be a collection of vectors. The Gram matrix of this collection is defined to be the nn-by-nn matrix whose entry in the ithi^\text{th} row and jthj^\text{th} column is aij=vivja_{ij} = v_i \cdot v_j, where \text{}\cdot denotes the dot product.

Consider the Gram matrix GG of the collection:

v1=(1,2,1)v2=(3,5,1)v3=(0,3,6)v4=(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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