I've been given the basis $B=\{(1,-1),(1,-2)\}$ and am asked to find the orthonormal basis using the Gram-Schmidt process. No problem! I have the orthogonal set of vectors $w_1 = (1,-1)$ and $w_2 = (\frac{-1}{2},\frac{-1}{2})$ meaning $||w_1|| = \frac{1}{\sqrt{2}} $ and $||w_2|| = \frac{1}{\sqrt{2}} $.
In making these into an orthonormal set of vectors however, I am to arrive at $\{\frac{1}{\sqrt{2}}(1,-1), \frac{1}{\sqrt{2}}(-1,-1)\}$. How is $(\frac{-1}{2},\frac{-1}{2})$ becoming $(-1,-1)$? Clearly I'm missing something obvious… Any help appreciated, thanks!
Best Answer
As Gerry Myerson points out, you divide by the norm to produce a unit vector. $\|w_1\|=\sqrt2$, not $1/\sqrt2$. When you divide $w_1$ by this you get $\frac1{\sqrt2}(1,-1)$ for the first vector as before. $\|w_2\|=1/\sqrt2$. Again, you divide $w_2$ by this value, giving $\sqrt2(-1/2,-1/2)=\frac1{\sqrt2}(-1,-1)$ for the second.