Specific non-singular matrix A and its relation to linearly independent vectors

Question

I am trying to prove the following:

The $m$ vectors $x_i$ are linearly independent if and only if the $m \times m$ matrix $A$ defined below is non-singular

$(i,j) \space element \space of \space A = \langle\ x_i,x_j \rangle\ $

I am confused by what this statement is saying, specifically what matrix A looks like. If we have a matrix where

$\begin{bmatrix}\langle\ x_1,x_1 \rangle\ & \langle\ x_1,x_2 \rangle\\ \langle\ x_2,x_1 \rangle\ & \langle\ x_2,x_2 \rangle\ \end{bmatrix}$

Is such a matrix even possible? I considered the one by one case

$\begin{bmatrix}\langle\ x_1,x_1 \rangle \end{bmatrix}$

This one by one matrix doesn't seem possible since whatever value is put into the first entry has to equal the square of itself.

I must be misunderstanding somehow. I considered the possibility that the question was asking to prove the following statement: the column or row vectors of a matrix are linearly independent if and only if the matrix is non-singular. Is this correct and should I go ahead and prove this statement instead? I suppose the notation throws me off.

Any insight or help would be greatly appreciated!

Answer 1

Not quite sure what do you mean by a matrix is possible since after all, you have written it down. It is alright to construct a matrix that is one by one of which the only entries is the norm of a vector.

The following might help you in your proof.

Note that $$\begin{bmatrix} \langle x_1, x_1 \rangle & \langle x_1, x_1 \rangle \\ \langle x_2, x_1 \rangle & \langle x_2, x_2 \rangle\end{bmatrix} =\begin{bmatrix} x_1^T \\ x_2^T\end{bmatrix}\begin{bmatrix} x_1 & x_2\end{bmatrix}$$