Basic Examples (3) 

The projection of one vector onto a 1D subspace:

In[1]:=

Out[1]=

This is the same as Mathematica’s Projection command, which projects one vector onto another:

In[2]:=

Out[2]=

The projection of a vector onto a subspace spanned by a list of vectors:

In[3]:=

Out[3]=

The projection using a weighted inner product:

In[4]:=

Out[4]=

Scope (3) 

Vectors with symbolic entries are assumed to represent complex numbers:

In[5]:=

Out[5]=

By using Dot as the inner product, Mathematica will assume all symbolic entries are real:

In[6]:=

Out[6]=

Project a vector in onto a subspace of :

In[7]:=

Out[7]=

Project a vector in onto a subspace of having an orthogonal basis:

In[8]:=

Out[8]=

Verify that the basis is orthonormal:

In[9]:=

Out[9]=

Since the basis is orthonormal, the projection is the sum of the projections of the vector onto the orthonormal basis vectors:

In[10]:=

Out[10]=

Projections in spaces of matrices (2) 

The projection of one matrix onto another using the standard matrix inner product:

In[11]:=

Out[11]=

---

The projection of a matrix onto a subspace of the space of 2×2 symmetric matrices using the standard matrix inner product:

In[12]:=

Out[12]=

Projections in function spaces (1) 

The projection of x² onto the subspace of trigonometric polynomials of degree 1 with the inner product Integrate[#1#2,{x,0,2π}]& gives the least-squares approximation to x² by a function in the subspace of C[0,2π] spanned by 1,Sin[x],Cos[x]:

In[13]:=

Out[13]=