= Basis = The '''bases''' for a linear space describe the space. Each member '''basis''' is independent. <> ---- == Description == Bases are independent vectors that can be linearly combined to reach every other vector in a linear space. If a basis vector is removed, the space necessarily shrinks. If the basis vectors are [[LinearAlgebra/Orthonormalization|orthonormalized]], they form an '''orthonormal basis'''. A convenient pair of orthonormal basis vectors in ''R^2^'' are ''[1 0]'' and ''[0 1]''. A convenient set of orthonormal basis vectors in ''R^3^'' are ''[1 0 0]'', ''[0 1 0]'', and ''[0 0 1]''. And so on. A [[LinearAlgebra/NullSpace|null space]] has no basis. All non-null spaces have infinitely many possible bases to choose from, because the ''only'' requirement on a basis is that it be independent. ---- == Change of Basis == A space can be linearly transformed to bring about a change of basis. This transformation can be expressed with a matrix. The inverse of that matrix then also expresses the inverse of the change of basis. Such a change of basis has a linear scaling effect on space. The scaling factor is the [[LinearAlgebra/Determinant|determinant]]. If the determinant is 0, then the matrix expresses a transformation that removes one (or more) basis vector(s). Such a transformation effectively collapses the space to a lower dimension. Any matrix that has basis is [[LinearAlgebra/Invertibility|invertible]], and therefore has a non-zero determinant. === Usage === If a matrix is [[LinearAlgebra/Diagonalization|diagonalizable]], identifying the change of basis that transforms it into a diagonal matrix enables several efficient strategies for solving systems. ---- CategoryRicottone