V=R^3 is an Inner product space and U=span {(1,2,1),(1,2,3)} is a subspace. I need to find an explicit formula for the orthogonal projection for Pu. I found an orthonormal base for U but I stucked with finding a formula for the projection.

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    $\begingroup$ If $(e_1,e_2)$ is an orthonormal basis for $U$, then the projection of $v\in V$ onto $U$ is given by $$\langle e_1,v\rangle e_1 + \langle e_2,v\rangle e_2.$$ $\endgroup$ – Math1000 Jun 2 '16 at 16:27
  • $\begingroup$ Why are you saying that $(1,2,3)\in \Bbb R^4$? $\endgroup$ – rschwieb Jun 2 '16 at 17:19
  • $\begingroup$ @rschwieb you right;, i changed it. $\endgroup$ – Roey Waitsman Jun 2 '16 at 17:34

If you want to find the orthogonal projection onto that plane, the strategy is simple:

  1. Find the unit normal $n$ to the plane. (Hint: cross product)
  2. Use the transformation that "removes" the component of a vector $v$ which is parallel to $n$ ($(v\cdot n)n$ is the part of $v$ parallel to $n$):

$$p(v)=v-(v\cdot n)n$$

Try your hand at these two steps.


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