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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. |
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.