Basis

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

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 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 R² space are [1 0] and [0 1].

Change of Basis

Any two independent vectors can form the basis for an R² 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 determinant.

Any matrix that has basis is 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 diagonalization. A matrix is transformed into a diagonal matrix of 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 = √(x² + y²), x = r*cosθ, and y = r*sinθ). The transformation of area is less so, and requires the Jacobian. Generically, the Jacobian is the determinant of the matrix describing the 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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