How to write the vector $(1,1,1)$ as a linear combination of an orthonormal basis.

324 Views Asked by At

As in the title I need to write the vector $v=(1,1,1)$ as a linear combination of the orthonormal basis

$$\left\{\left(\frac{1}{\sqrt{30}},\sqrt{\frac{5}{6}},\sqrt{\frac{2}{15}}\right),\left(\frac{1}{\sqrt{6}},-\frac{1}{\sqrt{6}},\sqrt{\frac{2}{3}}\right),\left(\frac{2}{\sqrt{5}},0,-\frac{1}{\sqrt{5}}\right)\right\}$$

How would I start go about doing this as I'm not sure where to begin.

1

There are 1 best solutions below

2
On

Given an orthonormal basis, $\left\{u_1, u_2, u_3 \right\}$, and a row vector $v$,

$$v=(vu_1^T) u_1+(vu_2^T) u_2+(vu_3^T) u_3$$