Rank Of Gram Matrix

Algebra Level 3

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 $G?$

4 1 2 3

1 solution

Otto Bretscher
Apr 14, 2016

Let $A$ be the $3\times 4$ matrix with $v_1^T,..,v_4^T$ as its columns. We see by inspection that $v_1,v_2,v_3$ are linearly independent, so that $rank(A)=3$ . Now the Gram Matrix is $A^T A$ and $rank(A^TA)=rank(A)$ so that the answer is $\boxed{3}$

How can you see by inspection that the vectors are linearly independent? Also how do you know by inspection that V4 is linearly dependent? Would you not have to put it into a matrix and row reduce? I don't see any combination of V1, V2, and V3 that equals V4.

Roadra Mojumder - 3 years, 11 months ago

Rank Of Gram Matrix

1 solution

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