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| == Bases == | == Description == |
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| 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. | 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. |
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| 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. | 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. |
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| A convenient pair of basis vectors for ''R^2^'' space are ''[1 0]'' and ''[0 1]''. | 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. |
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| Any two independent vectors can form the basis for an ''R^2^'' space, but ''[1 0]'' and ''[0 1]'' are the most convenient bases. The linear space can be transformed between bases, to normalize values into a convenient shape. | 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. |
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| This linear transformation can be expressed with a matrix; the inverse transformation can also be expressed with the inverse of that same matrix. | 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. |
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| The primary example of this is [[LinearAlgebra/Diagonalization|diagonalization]], where a matrix is linearly transformed to be a [[LinearAlgebra/SpecialMatrices#Diagonal_Matrices|diagonal matrix]] of [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvalues]]. | 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. 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.
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.