Linear dependence
Linear dependence is a property of a set of vectors or other linear objects in which at least one can be expressed as a linear combination of the others.
Geometry for Programmers
Linear dependence is a relationship in which one equation, vector, or other linear object can be expressed as a linear combination of others.
- How it is used
- Uses dependence to detect degenerate or unsolvable geometric systems before computing ray-triangle intersection parameters.
- How deep it goes
- Explains geometric consequences and implementation checks, but defers formal rank theory and floating-point error analysis.
- Examples
- Uses coincident hyperplanes, collinear interpolation points, degenerate triangles, parallel rays, and symbolic SymPy equations.
- What you can do after
- Diagnose geometric degeneracy by checking a system’s divisor before calculating intersection parameters.
Read how Geometry for Programmers explains linear dependence →
Math and Architectures of Deep Learning
A property of a set of vectors where at least one vector can be written as a linear combination of the others.
- How it is used
- Uses dependence to explain redundant equations, singular matrices, unsolvable systems, and eigenvector independence.
- How deep it goes
- Defines dependence algebraically, illustrates it geometrically, proves an eigenvector result, and connects it to determinants and inverses.
- Examples
- Uses collinear points, the vectors [1, 1] and [2, 2], and a two-equation redundant system.
- What you can do after
- Evaluate/understand dependence by finding nonzero scalar weights and recognizing its matrix and eigenvector consequences.
Read how Math and Architectures of Deep Learning explains linear dependence →
Related concepts
- weighted sumimplemented by · A weighted sum is an expression formed by multiplying terms by corresponding scalar coefficients and adding the resulting products.
- weighted averageimplemented by · A weighted average is an average in which each value contributes according to an assigned numerical weight.
- multicollinearityequivalent to · Multicollinearity is a condition in which two or more predictor variables are strongly linearly related, making their individual effects difficult to distinguish or estimate reliably.
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