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Projection (Linear Algebra)In linear algebra, a projection is a linear transformation P such that P2 = P, i.e., an idempotent transformation. A matrix is a projection if the transformation it represents is a projection. An m × m matrix projection maps an m-dimensional vector space onto a k-dimensional subspace (k ≤ m). A special class of projections is the class of orthogonal projections, which are self-adjoint projections. One such common projection is the projection of one vector in Rn onto another. For example, we can project the vector (1/2, 1/2)T onto the vector (0, 1)T, to get the vector (0, 1/2)T. We can describe in general the projection of one vector u onto another, v by -
where the dot represents the dot product. Since an inner product generalizes the idea of a dot product, then we have the equivalent formulation for any general inner product space: -
where <v1,v2> represents the inner product. This projection is indeed a projection, observe: -
by definition, then -
={\langle\mathbf{w},\alpha\mathbf{a}+\beta\mathbf{b}\rangle\over\langle\mathbf{w},\mathbf{w}\rangle}\mathbf{w} ={\langle\mathbf{w},\alpha\mathbf{a}\rangle\over\langle\mathbf{w},\mathbf{w}\rangle}\mathbf{w}+{\langle\mathbf{w},\beta\mathbf{b}\rangle\over\langle\mathbf{w},\mathbf{w}\rangle}\mathbf{w} -
=\alpha\,\mathrm{proj}_{\mathbf{w}}\,\mathbf{a}+\beta\,\mathrm{proj}_{\mathbf{w}}\,\mathbf{b} Projections (orthogonal and otherwise) play a major role in algorithms for certain linear algebra problems:
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