What is an intuitive explanation of a projection?

the orthogonal projection of a vector onto another vector.
Imagine a cart on a track that goes in a straight line. (Imagine also that the cart can go in either direction.) If b  b  is a vector that points in the direction of this track and a  a  is a vector representing a force you apply to the cart, then the projection of a  a  onto b is the part of that force that will determine the cart's acceleration along the track. (The rest of the force is orthogonal to the track, and won't contribute to that motion.)

Answer by James Brust:

Based on the "Linear Algebra" tag, I have two guesses as to what you mean by "projection".
Guess 1: You mean the orthogonal projection of a vector onto another vector.
Imagine a cart on a track that goes in a straight line. (Imagine also that the cart can go in either direction.) If [math]b[/math] is a vector that points in the direction of this track and [math]a[/math] is a vector representing a force you apply to the cart, then the projection of [math]a[/math] onto b is the part of that force that will determine the cart's acceleration along the track. (The rest of the force is orthogonal to the track, and won't contribute to that motion.)
Guess 2: You mean the general idea of a projection, an idempotent linear transformation from a vector space to itself.
If you think of what we would normally call a "projection" from [math]\mathbb{R}^n[/math] to one of its subspaces, each point is identified with a particular line that passes through it and intersects the target subspace somewhere, and we map the point to that intersection point. Clearly the points of this subspace have to be mapped to themselves. So it makes sense that applying the projection a second time shouldn't change the result. So, these projections are idempotent.

What is an intuitive explanation of a projection?

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