= Projections = When a vector does not exist in a column space, the '''projection''' is the best approximation of it in linear combinations of that column space. <> ---- == Vectors == Given vectors ''a'' and ''b'', ''a'' can be projected into ''C(b)'', the column space of ''b''. This projection ''p'' has an error term ''e''. === Trigonometric Approach === Projections with vectors can be calculated in terms of ''θ'' is the angle formed by ''a'' and ''b''. A vector in the direction of ''b'' with the magnitude of ''a'' is given by ''||b|| cos(θ)''. This can be called the '''scalar projection'''. However, a '''vector projection''' 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'', which can be calculated as ''a/||a||''. The projection vector is given by ''(||a|| cos(θ)) (a/||a||) = (||b|| cos(θ)) â''. === Algebraic Approach === Projections with vectors can also be calculated in terms of the vectors themselves, as they represent linear transformations. First, the [[LinearAlgebra/VectorMultiplication#Dot_Product|dot product]] can be substituted into the above formulas to give a scalar projection as ''a⋅b/||a||'' and a vector projection as ''(a⋅b/||a||) a/||a|| = (a⋅b/||a||) â''. The vector projection can then be reformulated like: ''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'' === Linear Algebraic Approach === The 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 [[LinearAlgebra/Orthogonality|orthogonal]] to ''e'', so ''a⋅(b - ax̂) = 0''. This simplifies to ''x̂ = (a⋅b)/(a⋅a)''. Altogether, the projection vector is ''p = a (a⋅b)/(a⋅a)''. The '''projection matrix''' '''''P''''' satisfies ''p = '''P'''b''. ''C('''P''')'', the column space of '''''P''''', is equivalent to ''C(a)''. It follows that '''''P''''' is also of [[LinearAlgebra/Rank|rank]] 1. === Properties === The projection matrix '''''P''''' is [[LinearAlgebra/MatrixProperties#Symmetry|symmetric]] (i.e. '''''P'''^T^ = '''P''''') and [[LinearAlgebra/MatrixProperties#Idempotency|idempotent]] (i.e. '''''P'''^2^ = '''P'''''). ---- == Matrices == Given a system as '''''A'''x = b'', if ''b'' is not in ''C('''A''')'', the column space of '''''A''''', then there is no possible solution for ''x''. The best approximation is expressed as '''''A'''x̂ = p'' where projection ''p'' estimates ''b'' with an error term ''e''. The error term can be characterized by ''e = b - p'' or ''e = b - '''A'''x̂''. ''e'' is orthogonal to ''R('''A''')'', the row space of '''''A'''''; equivalently it is orthogonal to ''C('''A'''^T^)''. Orthogonality in this context means that ''e'' is in the [[LinearAlgebra/NullSpaces|null space]], so '''''A'''^T^(b - '''A'''x̂) = 0''. The system of '''normal equations''' is '''''A'''^T^'''A'''x̂ = '''A'''^T^b''. This simplifies to ''x̂ = ('''A'''^T^'''A''')^-1^'''A'''^T^b''. Altogether, the projection is characterized by ''p = '''A'''('''A'''^T^'''A''')^-1^'''A'''^T^b''. The projection matrix '''''P''''' satisfies ''p = '''P'''b''. It is calculated as '''''P''' = '''A'''('''A'''^T^'''A''')^-1^'''A'''^T^''. ''b'' can also be projected onto ''e'', which geometrically means projecting into the null space of '''''A'''^T^''. Algebraically, if one projection matrix has been computed as '''''P''''', then the projection matrix for going the other way is ''('''I''' - '''P''')b''. === Properties === As above, the projection matrix '''''P''''' is symmetric and idempotent. If '''''A''''' is square, the above equations simplify rapidly. If ''b'' actually ''was'' in ''C('''A''')'', then '''''P''' = '''I'''''. Conversely, if ''b'' is orthogonal to ''C('''A''')'', then '''''P'''b = 0'' and ''b = e''. === Usage === [[Statistics/OrdinaryLeastSquares|This should look familiar.]] A projection is inherently the minimization of the error term. ---- CategoryRicottone