[Math] Why is it called “Orthogonal Projection”? Why not just “Projection”

linear algebra

Right now, we are learning decomposing vectors, but something I don't understand is the names given to this stuff

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For instance, in the text, the parallel component of y is said to be the orthogonal projection of y onto u. This makes no sense to me. Why is the word "orthogonal" even in there in the first place? I think I understand why they use "orthogonal" for z, but it makes no sense to me when they could just call it "orthogonal"

Best Answer

As Bill's answer explains, we can decompose every vector in the original space by using the projection map. This lends to an intuitive geometric interpretation of orthogonal projections.

If $p:V\to W$ is an $\color{Green}{orthogonal}$ projection down to a subspace, the fibers (pre-images) of every point $w\in W$ is perpendicular to the base (the subspace $W$). With $\dim V=2$ and $\dim W=1$:

$\hskip 0.4in$ orthogonal vs not orthogonal

The base (black line) and fibers (gray lines) can in general be viewed as higher-dimensional planes.