Linear system
A linear system is a set of linear relationships among variables, commonly expressed as simultaneous equations or a linear transformation, for which superposition applies.
Geometry for Programmers
A linear system is a collection of linear equations whose variables must satisfy all equations simultaneously.
- How it is used
- Frames geometric and engineering problems as simultaneous equations that can be visualized and solved.
- How deep it goes
- Develops definitions, hyperplane geometry, solution cases, matrix operations, solver choices, and a SymPy workflow.
- Examples
- Uses the two-trains intersection, ray–triangle intersection, and symbolic arc-to-line construction.
- What you can do after
- Apply it via SymPy or selected solver methods to formulate and solve small geometric problems.
Math and Architectures of Deep Learning
A set of linear equations considered together, typically represented in matrix form and solved for unknown variables.
- How it is used
- Models equations for solving unknown parameters in machine learning and linear algebra.
- How deep it goes
- Defines matrix form, exact and approximate solving, singularity, and SVD-based alternatives.
- Examples
- Uses toy training data, repeated equations, and overdetermined systems as concrete cases.
- What you can do after
- Apply pseudo-inverse or SVD to solve exact, approximate, or nonsquare systems.
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Related concepts
- singular value decompositionused by · Singular value decomposition is a matrix factorization that expresses a matrix as the product of orthogonal or unitary matrices and a diagonal matrix of nonnegative singular values.
- pseudo-inverseaddressed by · A pseudoinverse is a generalized matrix inverse that provides least-squares solutions when an ordinary inverse does not exist.
- Gaussian eliminationaddressed by · Gaussian elimination is a procedure that applies elementary row operations to transform a linear system or matrix into an equivalent triangular or row-reduced form.
- SymPyaddressed by · SymPy is an open-source Python library for symbolic mathematics and computer algebra.
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