Rank Of Gram Matrix

Algebra Level 5

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 \(\cdot\) denotes the dot product.

Consider the Gram matrix \(G\) of the collection: \[\begin{align} v_1 &= (1,2,1)\\ v_2 &= (-3,5,1)\\ v_3 &= (0,-3,6)\\ v_4 &= (4,-2,0). \end{align}\] What is the rank of \(G?\)

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