Basis
The bases for a linear space describe the space. Each member basis is independent.
Contents
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 orthonormalized, they form an orthonormal basis. A convenient pair of orthonormal basis vectors in R2 are [1 0] and [0 1]. A convenient set of orthonormal basis vectors in R3 are [1 0 0], [0 1 0], and [0 0 1]. And so on.
A 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 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 invertible, and therefore has a non-zero determinant.
Usage
If a matrix is diagonalizable, identifying the change of basis that transforms it into a diagonal matrix enables several efficient strategies for solving systems.
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 trick is that space was also transformed in the change of basis. 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.