Let {E1,E2,...En} be an orthogonal basis of Rn. Given k, 1<=k<=n, define Pk: Rn -> Rn by [itex] P_{k} (r_{1} E_{1} + ... + r_{n} E_{n}) = r_{k} E_{k}. [/itex] Show that [itex] P_{k} = proj_{U} () [/itex] where U = span {Ek}(adsbygoogle = window.adsbygoogle || []).push({});

well [tex] \mbox{proj}_{U} \vec{m}= \sum_{i} \frac{ m \bullet u_{i}}{||u_{i}||^2} \vec{u}

[/tex]

right?

here we have Pk transforming linear combination of the orthogonal basis into rk Ek same index as the subscript of P

would it turn into

[tex] \mbox{proj}_{U} \vec{m}= \frac{ m \bullet E_{1}}{||E_{1}||^2}\vec{E_{1}} + ... + \frac{ m \bullet E_{n}}{||E_{n}||^2}\vec{E_{n}} [/tex]

and the whole stuff in front of each Ei can be interpreted as the Ri, a scalar multiple yes?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Linear transformations and orthogonal basis

**Physics Forums | Science Articles, Homework Help, Discussion**