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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:
Here we provide two proofs. The first [2] operates in the general case, using linear maps. The second proof [6] looks at the homogeneous system =, where is a with rank, and shows explicitly that there exists a set of linearly independent solutions that span the null space of .
In linear algebra, two n-by-n matrices A and B are called similar if there exists an invertible n-by-n matrix P such that =. Similar matrices represent the same linear map under two (possibly) different bases, with P being the change-of-basis matrix.
In mathematics, particularly in algebra, an indeterminate system is a system of simultaneous equations (e.g., linear equations) which has more than one solution (sometimes infinitely many solutions). [1] In the case of a linear system, the system may be said to be underspecified, in which case the presence of more than one solution would imply ...
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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English: Linear Algebra by Jim Hefferon, along with its answers to exercises, is a text for a first undergraduate course. It is Free. Use it as the main book, as a supplement, or for independent study.
The homogeneous (with all constant terms equal to zero) underdetermined linear system always has non-trivial solutions (in addition to the trivial solution where all the unknowns are zero). There are an infinity of such solutions, which form a vector space , whose dimension is the difference between the number of unknowns and the rank of the ...
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