Linear Projection

A Linear Projection is a linear transformation that maps vectors from a vector space to itself while satisfying the idempotence property (where applying it twice yields the same result as applying it once).

• Context:
  • It can typically preserve Linear Subspaces by mapping vectors onto linear subspaces.
  • It can typically satisfy Idempotence where P² = P for any linear projection operator P.
  • It can typically decompose Vector Spaces into direct sums of its image space and kernel space.
  • It can typically maintain Linear Combinations when applied to vector sums and scalar multiplications.
  • It can typically leave its Image Space unchanged under repeated linear projection applications.
  • ...
  • It can often reduce Dimensionality by projecting high-dimensional vectors onto lower-dimensional linear subspaces.
  • It can often facilitate Geometric Visualization through coordinate transformations.
  • It can often enable Least Squares Approximations in statistical analysis and data fitting.
  • It can often support Feature Extraction in machine learning algorithms.
  • ...
  • It can range from being a Simple Linear Projection to being a Complex Linear Projection, depending on its linear projection computational complexity.
  • It can range from being a Low-Rank Linear Projection to being a Full-Rank Linear Projection, depending on its linear projection rank.
  • It can range from being a Sparse Linear Projection to being a Dense Linear Projection, depending on its linear projection matrix density.
  • ...
  • It can be represented by Projection Matrices satisfying P² = P.
  • It can be characterized by Eigenvalues that are either 0 or 1.
  • It can be implemented through Matrix Multiplication operations.
  • It can be optimized using Numerical Linear Algebra techniques.
  • ...
• Examples:
  • Geometric Linear Projections, such as:
    • Orthogonal Projection, which minimizes distance to the linear projection subspace.
    • Oblique Linear Projection, which projects along non-perpendicular directions.
    • Perspective Linear Projection, used in computer graphics for 3D rendering.
  • Algebraic Linear Projections, such as:
    • Spectral Linear Projection, decomposing along eigenvectors.
    • Canonical Linear Projection, mapping to coordinate axes.
    • Component Linear Projection, extracting specific vector components.
  • Application-Specific Linear Projections, such as:
    • Principal Component Linear Projection in dimensionality reduction.
    • Least Squares Linear Projection in regression analysis.
    • Hilbert Space Projection in functional analysis.
  • Machine Learning Linear Projections, such as:
    • Random Linear Projection for dimensionality reduction.
    • Learned Linear Projection in neural network layers.
    • Attention Linear Projection in transformer architectures.
  • ...
• Counter-Examples:
  • Nonlinear Transformation, which lacks the linearity property required for linear projections.
  • Linear Isomorphism, which is bijective and thus not idempotent like linear projections.
  • General Linear Transformation, which may not satisfy P² = P required for linear projections.
  • Affine Transformation, which includes translations not present in linear projections.
• See: Matrix (Mathematics), Orthogonal Projection, Linear Transformation Operation, Vector Space, Idempotent Operator, Projection Matrix, Hilbert Space Operator, Eigenspace, Direct Sum Decomposition.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Projection_(linear_algebra)#Orthogonal_projections Retrieved:2015-2-2.
  • In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that . That is, whenever P is applied twice to any value, it gives the same result as if it were applied once (idempotent). It leaves its image unchanged. Though abstract, this definition of "projection" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object.

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