= Basis =

The '''bases''' for a linear space describe the space. Each member '''basis''' is independent.

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== Bases ==

For any linear space, the bases are independent vectors that can be linearly combined to reach every other vector in the space. If a basis is removed, the space necessarily shrinks.

A [[LinearAlgebra/NullSpaces|null space]] has no basis, but all other spaces have infinitely many possible bases, because the ''only'' requirement on a basis is that it be independent.

A convenient pair of basis vectors for ''R^2^'' space are ''[1 0]'' and ''[0 1]''.

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== Change of Basis ==

Any two independent vectors can form the basis for an ''R^2^'' space, but ''[1 0]'' and ''[0 1]'' are the most convenient bases. A linear space can be linearly transformed to effect a change of basis.

This linear transformation can be expressed with a matrix; the inverse transformation (to return to the old basis) can is the inverse of that same matrix.



=== Determinants ===

A change of basis has a linear scaling effect on space. The scaling factor is the [[LinearAlgebra/Determinants|determinant]].

Any matrix that has basis is [[LinearAlgebra/MatrixProperties#Invertible|invertible]], and ergo has a non-zero determinant.



=== Diagonalization ===

The primary example of how a change of basis can be used to ease solutions is [[LinearAlgebra/Diagonalization|diagonalization]]. A matrix is transformed into a [[LinearAlgebra/SpecialMatrices#Diagonal_Matrices|diagonal matrix]] of [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvalues]]. Many powerful rules for evaluation apply to diagonal matrices.



=== Jacobians ===

Some differentiation problems are more easily solved in polar coordinates than in Cartesian coordinates. The transformation of points is simple (i.e., ''r = √(x^2^ + y^2^)'', ''x = r*cosθ'', and ''y = r*sinθ''). The transformation of area is less so, and requires the '''Jacobian'''. Generically, the Jacobian is the [[LinearAlgebra/Determinants|determinant]] of the matrix describing the [[Calculus/ChainRule|chain rule]] operations necessary.

{{{
    | ∂x ∂x |
    | ―― ―― |
    | ∂u ∂v |   ∂x ∂y   ∂x ∂y
det |       | = ―― ―― - ―― ――
    | ∂y ∂y |   ∂u ∂v   ∂v ∂u
    | ―― ―― |
    | ∂u ∂v |
}}}

Concretely, for the transformation of 2-dimensional polar coordinates to 2-dimensional Cartesian coordinates, the Jacobian is:

{{{
    | ∂x ∂x |       
    | ―― ―― |
    | ∂θ ∂r |       | cosθ -r*sinθ |
det |       | = det | sinθ  r*cosθ | = (cosθ)(r*cosθ) - (-r*sinθ)(sinθ) = r
    | ∂y ∂y |
    | ―― ―― |
    | ∂θ ∂r |
}}}

Therefore ''dxdy = rdrdθ''.



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