= 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 span an entire subspace. If a basis vector is removed, the subspace 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. ---- == Coordinatization == Given a space (like ''R^n^'') and bases that span that space (the set ''B = u,,1,, ... u,,n,,''), any vector ''v'' in that space can be represented as a linear combination of the bases. In other words, there must be a set of coefficients ''c,,1,, ... c,,n,,'' that satisfy: ''c,,1,,u,,1,, + ... + c,,n,,u,,n,, = v'' The vector of coefficients (''v,,B,, = [c,,1,,, ... c,,n,,]'') is called the '''coordinate vector''' relative to ''B''. By rewriting the set of bases as a matrix '''''M''',,B,,'', the above statement becomes: '''''M''',,B,, v,,B,, = v'' ''v,,B,,'' can then be identified through [[LinearAlgebra/Elimination|elimination]] of the augmented matrix ''[u,,1,, ... u,,n,, | v]'', or through most software packages as '''''M''',,B,, \ v''. ---- == Change of Basis == A space can be [[LinearAlgebra/LinearMapping|linearly transformed]] to bring about a change of basis. This transformation can be expressed with [[LinearAlgebra/MatrixMultiplication|matrix multiplication]]. Following from the above notation, since the following are known to be true: '''''M''',,B',, v,,B',, = v'' '''''M''',,B,, v,,B,, = v'' It must also be true that: '''''M''',,B',, v,,B',, = '''M''',,B,, v,,B,,'' ''v,,B',, = '''M''',,B',,^-1^ '''M''',,B,, v,,B,,'' The change of basis matrix '''''C''',,B,B',,'' (note the subscript, indicating that it transforms from ''B'' to ''B'``'') is defined as: '''''C''',,B,B',, = '''M''',,B',,^-1^ '''M''',,B,,'' This can then be identified through [[LinearAlgebra/Elimination|elimination]] of the augmented matrix ''['''M''',,B',, | '''M''',,B,,]'', or through most software packages as '''''M''',,B',, \ '''M''',,B,,''. 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