Similarity

Change-of-Basis

Let α = [e₁, e₂, … , en] and β = [b₁, b₂, … , bn] be two ordered bases of n-dimensional vector space V and let T : V ⇾ V be a linear transformation. Then any vector v can be expanded in uniae way with respect to each basis:
\begin{align*} \mathbf{v} &= x_1 {\bf e}_1 +x_2 {\bf e}_2 + \cdots + x_n {\bf e}_n \\ &= y_1 {\bf b}_1 + y_2 {\bf b}_2 + \cdots + y_n {\bf b}_n . \end{align*}

Properties of similar transformations

   

 

  1. Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International
  2. Beezer, R.A., A First Course in Linear Algebra, 2017.
  3. Fitzpatrick, S., Linear Algebra: A second course, featuring proofs and Python. 2023.