# Linear Algebra - Understanding how to use the Gram Schmidt Orthogonalization Procedure [duplicate]

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Use the Gram Schmidt Orthogonalization Procedure to transform the basis $\{(1,2,1), (1,0,1), (3,1,0)\}$ into an orthogonal basis for $R^4$.

I haven't been able to understand how to use the Gram Schmidt Orthogonalization Procedure properly, I just couldn't grasp it during my lecture. If anyone could explain how to do this, that would be awesome!

## marked as duplicate by davidlowryduda♦Sep 21 '15 at 1:09

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• Here's an excellent visual guide for what's going on: en.wikipedia.org/wiki/…. You should watch it as you are trying to do it. – Robert Wolfe Feb 11 '14 at 20:27
• Well, you're sorta in trouble, cause none of those vectors are in $\mathbb{R}^4$ - they are in $\mathbb{R}^3$ (but do form a basis for $\mathbb{R}^3$). The essential idea is that given vectors $a,b$, you can split $b$ into a vector parallel to $a$, $b^{||}$ and a vector perpendicular to $a$, $b^\perp$. Gram schmidt greedily orthogonalizes by removing the parallel parts along the previous vectors you've already processed. – Batman Feb 11 '14 at 20:36