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!
Best Answer
Gram-Schmidt can be used to reduce down to a linearly independent basis! Normally, the way it works is you find a vector orthogonal to the vectors so far, then normalise it. This orthogonal vector will be $0$ if and only if the vector was dependent on the previous ones. So, if you get a zero, ignore that vector and move on to the next one.
It works pretty simply, if you understand Gram-Schmidt geometrically. What Gram-Schmidt does is, at stage $j$, project $v_j$ onto the span of $v_1, \ldots v_{j-1}$, and take the orthogonal complement of this projection. When this returns a zero, this means that the projection onto the span coincides with $v_j$ in the first place, which can only happen when $v_j$ belongs in the span to begin with. Thus, removing $v_j$ from the span will not reduce it.