Vector Operations

 v[[i]] or Part[v,i] give the i element in the vector v c v scalar multiplication of c times the vector v u.v dot product of two vectors Norm[v] give the norm of v Normalize[v] give a unit vector in the direction of v Standardize[v] shift v to have zero mean and unit sample variance Standardize[v,f1] shift v by f1[v] and scale to have unit sample variance

Basic vector operations.

This is a vector in three dimensions:
 In:= Out= This gives a vector u in the direction opposite to v with twice the magnitude:
 In:= Out= This reassigns the first component of u to be its negative:
 In:= Out= This gives the dot product of u and v:
 In:= Out= This is the norm of v:
 In:= Out= This is the unit vector in the same direction as v:
 In:= Out= This verifies that the norm is 1:
 In:= Out= Transform v to have zero mean and unit sample variance:
 In:= Out= This shows the transformed values have mean 0 and variance 1:
 In:= Out= Two vectors are orthogonal if their dot product is zero. A set of vectors is orthonormal if they are all unit vectors and are pairwise orthogonal.

 Projection[u,v] give the orthogonal projection of u onto v Orthogonalize[{v1,v2,…}] generate an orthonormal set from the given list of vectors

Orthogonal vector operations.

This gives the projection of u onto v:
 In:= Out= p is a scalar multiple of v:
 In:= Out= u-p is orthogonal to v:
 In:= Out= Starting from the set of vectors {u,v}, this finds an orthonormal set of two vectors:
 In:= Out= When one of the vectors is linearly dependent on the vectors preceding it, the corresponding position in the result will be a zero vector:
 In:= Out= 