Question

show that the orthogonal projection operator,
PROJw : V -> W is a linear map

Answer 1

Lets start off with a specific example. A three-dimensional vector space contains subspace V, a one-dimensional vector space, and subspace W, a two dimensional vector space. V is the set of points on the line x = 2t, y = 3t, z = 6t, and W is the plane x + 2y + z = 0. Draw a line through
P(2t, 3t, 6t) and orthogonal to W at point P', let r be the vector from the origin O to point P, and r' be the vector from O to P'. The projection maps r to r'.
Suppose t takes on the values t(1) and t(2). Let r(i) = <2t(i), 3t(i), 6t(i)> and let r'(i) be the projection of r(i) onto W, i = 1, 2. If we can show that r(1) + r(2) maps to r'(1) + r'(2) and for all real numbers a, a*r(1) maps to a*r'(1), then we have proven that the projection is a linear map.
I chose this example because it is easier to visualize. If we think about this from a geometric stance, what we have here is a plane W and a line V that omtersecte W at O. There is exactly one plane U containing V and orthogonal to W. Planes U and W intersect at line V', so the projection maps line V to line V'.

Answer 2

intersects W, not omtersecte W.