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| == 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''. ---- |
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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,,''. |
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| 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 trick is that space was also transformed in the change of basis. The '''Jacobian''' is the [[LinearAlgebra/Determinant|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''. |
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.
Coordinatization
Given a space (like Rn) and bases that span that space (the set B = u1 ... un), 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 c1 ... cn that satisfy:
c1u1 + ... + cnun = v
The vector of coefficients (vB = [c1, ... cn]) is called the coordinate vector relative to B.
By rewriting the set of bases as a matrix MB, the above statement becomes:
MB vB = v
vB can then be identified through elimination of the augmented matrix [u1 ... un | v], or through most software packages as MB \ v.
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.
Following from the above notation, since the following are known to be true:
MB' vB' = v
MB vB = v
It must also be true that:
MB' vB' = MB vB
vB' = MB'-1 MB vB
The change of basis matrix CB,B' (note the subscript indicating that it transforms from B to B') is defined as:
CB,B' = MB'-1 MB
This can then be identified through elimination of the augmented matrix [MB' | MB], or through most software packages as MB' \ MB.
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.