Consider inner product space $\mathbb{R}^4$ with respect to the dot product.
Find an orthonormal basis for the subspace $W$ of $\mathbb{R}^4$ generated by:
$\left\lbrace \begin{bmatrix} 4\\2\\6\\-2 \end{bmatrix}, \begin{bmatrix} 1\\-1\\3\\-1 \end{bmatrix}, \begin{bmatrix} 1\\2\\0\\0 \end{bmatrix}, \begin{bmatrix} 1\\5\\-3\\1 \end{bmatrix} \right\rbrace $
Ok so I tried using Gram-Schmidt, however, these vectors are linearly dependent. I've read somewhere that Gram-Schmidt can only be applied to a set of linearly independent vectors..is this true? I end up getting two zero vectors in my set of orthogonal vectors! Is this against the rules? Please help!
Please be very elaborate as I have searched everywhere for help!