Wolfram Function Repository
Instantuse addon functions for the Wolfram Language
Function Repository Resource:
Find the coordinate vector of a vector with respect to a basis
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. 
Get the coordinates of a given vector in a given basis:
In[1]:= 

Out[1]= 

When ℝ^{1} is viewed as a vector space, its elements are singletons, i.e. (α). The standard basis for ℝ^{1} is {(1)}, so to find the coordinates of the vector (5) with respect to the basis {(1)}, proceed as follows:
In[2]:= 

Out[2]= 

Find the coordinates of a vector in ℝ^{4} with respect to a basis of ℝ^{4}:
In[3]:= 

Out[4]= 

Verify the result:
In[5]:= 

Out[5]= 

Find the coordinates of a vector in ℝ^{4} with respect to a basis for a threedimensional subspace of ℝ^{4}:
In[6]:= 

Out[7]= 

Verify the result:
In[8]:= 

Out[8]= 

Find the coordinates of a list of vectors with respect to a basis:
In[9]:= 

Out[10]= 

Verify the result:
In[11]:= 

Out[11]= 

Find the coordinate vector of a vector with symbolic entries:
In[12]:= 

Out[12]= 

Verify the result:
In[13]:= 

Out[13]= 

If the vectors listed as the basis vectors are not independent, an error message is returned:
In[14]:= 

Out[14]= 

Check that the given list of vectors is not independent:
In[15]:= 

Out[15]= 

If a vector is not in the span of the given list of vectors, an error message is returned:
In[16]:= 

Out[16]= 

Check that at least one of the given vectors is not in the span:
In[17]:= 

Out[17]= 

The standard basis for the space of 2×4 matrices is:
In[18]:= 

Out[18]= 

Find the coordinate vector of the 2×4 matrix with respect to this basis:
In[19]:= 

Out[19]= 

Verify the result:
In[20]:= 

Out[20]= 

Find the coordinates of a matrix with symbolic entries:
In[21]:= 

Out[19]= 

Check that the result is correct:
In[22]:= 

Out[22]= 

Find the coordinates of a list of two vectors (functions) with respect to the standard basis {1,x,x^{2},x^{3}} of the space of polynomials of degree at most 3:
In[23]:= 

Out[19]= 

Verify the result:
In[24]:= 

Out[24]= 

Find the coordinates of two vectors (functions) with respect to a basis for a twodimensional subspace of the space of polynomials of degree at most 4:
In[25]:= 

Out[19]= 

Verify the result:
In[26]:= 

Out[26]= 

This example exploits trigonometric identities to find the vector's coordinates:
In[27]:= 

Out[19]= 

Verify the result:
In[28]:= 

Out[28]= 

Example involving exponential and hyperbolic functions:
In[29]:= 

Out[19]= 

Verify the result:
In[30]:= 

Out[30]= 

A more complicated function:
In[31]:= 

Out[31]= 

Verify the result:
In[32]:= 

Out[32]= 

The following example involves a dependent list of exponential and hyperbolic functions. The given list of vectors is dependent because the vector cos^{2}(x) is a linear combination of 1 and cos(2x):
In[33]:= 

Out[32]= 

This work is licensed under a Creative Commons Attribution 4.0 International License