Geometry and algebra of linear maps, vectors, and matrices
A projection answers "what is the closest point to b that lives in a given subspace?" Picture a point floating above a floor: its projection is the spot on the floor directly below it, the foot of the perpendicular. It is the best approximation of b available within the subspace.
To project a vector b onto a single direction a, scale a by how much of b lies along it (a dot product), normalized by a's own length-squared:
Drag b around the figure and watch its shadow slide along the line a, always landing at the closest point, with the dashed error segment meeting the line at a right angle.