Function Repository Resource:

CoordinateVector

Find the coordinate vector of a vector with respect to a basis

Contributed by: Dennis M Schneider

ResourceFunction["CoordinateVector"][vec,basis]

finds the coordinates of the vector vec with respect to the basis basis.

ResourceFunction["CoordinateVector"][vec,basis,var]

finds the coordinates of a vector when the vector and the basis vectors are members of a function space consisting of functions of var.

Details and Options

The input vector is given in terms of the standard basis.

The input vector may be a list of vectors.

Examples

Basic Examples (1)

Get the coordinates of a given vector in a given basis:

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Scope (15)

Vectors in ℝⁿ

When ℝ¹ is viewed as a vector space, its elements are singletons, i.e. (α). The standard basis for ℝ¹ is {(1)}, so to find the coordinates of the vector (5) with respect to the basis {(1)}, proceed as follows:

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Find the coordinates of a vector in ℝ⁴ with respect to a basis of ℝ⁴:

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Verify the result:

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Find the coordinates of a vector in ℝ⁴ with respect to a basis for a three-dimensional subspace of ℝ⁴:

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Verify the result:

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Find the coordinates of a list of vectors with respect to a basis:

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Verify the result:

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Find the coordinate vector of a vector with symbolic entries:

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Verify the result:

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If the vectors listed as the basis vectors are not independent, an error message is returned:

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Check that the given list of vectors is not independent:

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If a vector is not in the span of the given list of vectors, an error message is returned:

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Check that at least one of the given vectors is not in the span:

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Spaces of matrices (2)

The standard basis for the space of 2×4 matrices is:

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Find the coordinate vector of the 2×4 matrix with respect to this basis:

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Verify the result:

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Find the coordinates of a matrix with symbolic entries:

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Check that the result is correct:

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Function spaces (6)

Find the coordinates of a list of two vectors (functions) with respect to the standard basis {1,x,x²,x³} of the space of polynomials of degree at most 3:

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$basis = {1, x, x^2, x^3}; coords = ResourceFunction[ "CoordinateVector"][{1 - 2 x + 3 x^2, 2 x - x^2 + x^3}, basis, x]$

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Verify the result:

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Find the coordinates of two vectors (functions) with respect to a basis for a two-dimensional subspace of the space of polynomials of degree at most 4:

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$basis = {-x + x^3, -x^2 + x^4}; coords = ResourceFunction[ "CoordinateVector"][{2 (-x + x^3) + 3 (-x^2 + x^4), x (x^2 - 1) (x + 3)}, basis, x]$

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Verify the result:

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$coords.basis == {2 (-x + x^3) + 3 (-x^2 + x^4), x (x^2 - 1) (x + 3)} // Simplify$

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This example exploits trigonometric identities to find the vector's coordinates:

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$basis = {1, Cos[2 t]}; coords = ResourceFunction["CoordinateVector"][{Cos[t]^2, Sin[t]^2}, basis, t]$

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Verify the result:

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Example involving exponential and hyperbolic functions:

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$basis = {1, Cos[x], Cos[x]^2, Exp[x], Exp[-x]}; coords = ResourceFunction[ "CoordinateVector"][{1, Cos[x], Cos[2 x], Sinh[x]}, basis, x]$

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Verify the result:

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A more complicated function:

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$basis = {Exp[2 x], x Exp[2 x], x^2 Exp[2 x]}; coords = ResourceFunction["CoordinateVector"][ D[5 Exp[2 x] + 2 x Exp[2 x] - x^2 Exp[2 x], x], {Exp[2 x], x Exp[2 x], x^2 Exp[2 x]}, x]$

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Verify the result:

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The following example involves a dependent list of exponential and hyperbolic functions. The given list of vectors is dependent because the vector cos²(x) is a linear combination of 1 and cos(2x):

In[33]:=

$basis = {1, Cos[x], Cos[2 x], Sinh[x], Cosh[x], Cos[x]^2}; coords = ResourceFunction["CoordinateVector"][{Exp[x], Exp[-x]}, basis, x]$