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In three-dimensional Euclidean space, these three planes represent solutions to linear equations, and their intersection represents the set of common solutions: in this case, a unique point. The blue line is the common solution to two of these equations. Linear algebra is the branch of mathematics concerning linear equations such as:
Formally, the construction is as follows. [1] Let be a vector space over a field, and let be a subspace of .We define an equivalence relation on by stating that if .That is, is related to if and only if one can be obtained from the other by adding an element of .
Because a solution to a linear system must satisfy all of the equations, the solution set is the intersection of these lines, and is hence either a line, a single point, or the empty set. For three variables, each linear equation determines a plane in three-dimensional space , and the solution set is the intersection of these planes.
The system + =, + = has exactly one solution: x = 1, y = 2 The nonlinear system + =, + = has the two solutions (x, y) = (1, 0) and (x, y) = (0, 1), while + + =, + + =, + + = has an infinite number of solutions because the third equation is the first equation plus twice the second one and hence contains no independent information; thus any value of z can be chosen and values of x and y can be ...
Fredholm's theorem (linear algebra) Freidlin–Wentzell theorem (stochastic processes) Freiman's theorem (number theory) Freudenthal suspension theorem (homotopy theory) Freyd's adjoint functor theorem (category theory) Friedberg–Muchnik theorem (mathematical logic) Friendship theorem (graph theory) Frobenius determinant theorem (group theory)
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In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the ...
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