This article explains the intuition behind the change of basis matrix.
Change of basis
You can represent a vector →v by its coordinates in a basis. For instance, here are the coordinates for the vector →v in two different basis (e)=(e1,e2) and (f)=(f1,f2):
Base 1 (e1, e2)
→v=(−10)e=−1e1+0e2
Base 2 (f1, f2)
→v=(−0.5+0.5)f=−0.5f1+0.5f2
The change of basis matrix
We can use a matrix to transform the coordinates of →v in the basis (e) into the coordinates in the basis (f). This matrix is called the change of basis matrix.
How do we find this matrix? Since it takes the coordinates of a vector in basis (e) and gives the coordinates in the basis (f), we can start by expressing the vectors e1 and e2 in basis (f).
- In basis (e): e1=(10)e and e2=(01)e
…and…
- In basis (f): e1=(0.5−0.5)f and e2=(0.50.5)f.
So we are looking for the matrix Mf←e such that:
- (0.5−0.5)f=Mf←e⋅(10)e
…and…
- (0.50.5)f=Mf←e⋅(01)e
Since Mf←e⋅(1,0)⊤ is the first column of Mf←e and Mf←e⋅(0,1)⊤ is the second column of Mf←e, we find that the change of basis matrix is the matrix made of the vector column of the old basis (e) expressed in the new basis (f). We note →vf the coordinates of vector →v in base (f).
- Mf←e=[→e1f→e2f]=(0.50.5−0.50.5)
Using this matrix, we can translate the coordinates for →v from basis (e) to basis (f):
→v:(−0.5+0.5)f=Mf←e⋅(−10)eFrom vector notation to sum notation
Sometimes, it’s useful to remember that matrix multiplication can be expressed as a sum of vectors using linearity.
The change of basis matrix allows us to write equations where some vectors are expressed in base (f) (left hand side) while others are in base (e) (right hand side).
Using the notation →vf for the coordinates of vector →v in base (f):
(ab)f=Mf←e⋅(cd)e a→f1f+b→f2f=Mf←e⋅(c→e1e+d→e2e) a→f1f+b→f2f=c(Mf←e⋅→e1e)+d(Mf←e⋅→e2e) a→f1f+b→f2f=c→e1f+d→e2fWhere every vector in the last equation is expressed in base (f).