inner product of vectors, dot products, orthogonality

$u \dot v = u^T v$

$u \dot v = v \dot u $

length (or norm) of vector $v$ is the square root of its inner product. This can be seen from the v [a,b], whose length(norm) is sqrt(a^2 + b^2)

u \dot v = ||u|| ||v|| cos \theta
||u-v|| = ||u||^2 + ||v||^2 - 2||u|| ||v|| cos\theta

Two vector $u$ and $v$ are orthogonal if and only if $u \dot v = 0$. 

U has orthonormal columns if and only if U^T U = I

orthogonal projection of vector y to u:
y^hat = \frac{y \dot u}{u \dot u } u

Orthogonal projection of a point y to W space with {u1, u2, ... up} basis can be found by orthogonal projections on each base vector, u1, u2, ..., u_p.