= Projection = A '''projection''' is an approximation within a column space. <> ---- == Description == Given vectors ''a'' and ''b'', ''a'' can be projected into the column space of ''b'' (i.e., ''C(b)''). This projection ''p'' has an error term ''e''. If ''a'' is [[LinearAlgebra/Orthogonality|orthogonal]] to ''C(b)'', there is no projection. ---- == Scalar Projection == A vector in the direction of ''b'' with the magnitude of ''a'' is given by ''||b|| cos(θ)'' where ''θ'' is the angle formed by ''a'' and ''b''. This can be called the '''scalar projection'''. The [[Calculus/VectorOperations#Dot_Product|dot product]] can be substituted into this definition to give ''a⋅b/||a||'' or ''a^T^b/||a||''. ---- == Vector Projection == A '''vector projection''' is very similar to the scalar projection, but should have a magnitude based on how much ''a'' moved through ''C(b)''. This is captured by ''â'', the unit vector in the direction of ''a'', and is calculated as ''a/||a||''. Altogether, the vector projection could be given by any of: * ''||b|| cos(θ) â'' * ''(a⋅b/||a||) â'' * ''(a^T^b/||a||) â'' Generally these reformulations are more useful: ''p = (a⋅b/||a||) a/||a||'' ''p = (a⋅b/||a||^2^) a'' ''p = (a⋅b/a⋅a) a'' ...or: ''p = (a⋅b/||a||) â'' ''p = (â⋅b) â'' The [[LinearAlgebra|linear algebra]] view of this is that linear transformation from vector ''a'' to projection vector ''p'' is expressed as ''p = ax̂''. The projection carries an error term that can be characterized by ''e = b - p'' or ''e = b - ax̂''. ''a'' is [[Calculus/Orthogonality|orthogonal]] to ''e'', so ''a⋅(b - ax̂) = 0''. This simplifies to ''x̂ = (a⋅b)/(a⋅a)''. Again, the vector projection is given by ''p = a (a⋅b)/(a⋅a)''. ---- CategoryRicottone