Very straightforward question. I have read time and again in my book that it is independent but I don't understand why?
Wouldn't changing the basis mean changing the length of the projection?
2026-03-26 16:10:13.1774541413
Bumbble Comm
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Why is the projection of a vector V onto a span W, independent of the orthogonal basis of W.
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I will assume that when you say "projection", it is expressed as a vector of coordinates relative to the basis.
It would not change the length, as you only consider orthogonal bases. By definition, orthogonal bases consist of vectors of length $1$, so the coordinates relative to all bases will have the same length.
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If I take the leaning tower of Pisa, and hang a weight from the top to the ground, then I'm projecting it orthogonally into the plane of the ground. The point that that weight touches the ground is the same whether I choose to use a pair of basis vectors where one is a meter pointing north and the other is a meter pointing west or if I choose to use a pair of basis vectors where one is foot in NW and the other is a foot NE.
The physical place that the projection reaches is the same. The coefficients of those basis vectors is different. I think you're getting confused and thinking that the coefficients give the length.