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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. A linear space can be linearly transformed to effect a 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. |
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| 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. | 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. Any matrix that has basis is [[LinearAlgebra/Invertibility|invertible]], and therefore has a non-zero determinant. |
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| === Determinants === | === Usage === |
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| A change of basis has a linear scaling effect on space. The scaling factor is the [[LinearAlgebra/Determinants|determinant]]. | 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. |
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| 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. |
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. |
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.