Canonical decomposition of a linear transformation

Dependencies:

1. Basis of F^n
2. Coordinatization over a basis
3. Basis change is an isomorphic linear transformation
4. Vector spaces are isomorphic iff their dimensions are same
5. Composition of linear transformations
6. Vector space isomorphism is an equivalence relation

Let $T: U \mapsto V$ be a linear transformation. Let $P = [u_1, u_2, \ldots, u_m]$ be a basis of $U$ and $Q = [v_1, v_2, \ldots, v_n]$ be a basis of $V$.

Then $T = T_3T_2T_1$ where $T_1, T_2, T_3$ are linear transformations such that:

• $T_1: U \mapsto F^m$ where $T_1(u) = [u]_P$.
• $T_2: F^m \mapsto F^n$
• $T_3: F^n \mapsto V$ where $T_3([a_1, a_2, \ldots, a_n]) = \sum_{i=1}^n a_iv_i$.

If $U = V$ and $P = Q$, then $T_1 = T_3^{-1}$.

Proof

Since $T_1$ and $T_3$ are basis-changers over vector spaces with the same dimension, they are isomorphic linear transformations.

When $U = V$ and $u_i = v_i$, $T_1$ is an inverse of $T_3$.

Define $T_2 = T_3^{-1}TT_1^{-1}$. Since inverse of an isomorphism is also an isomorphism and since composition of linear transformations is a linear transformation, $T_2$ is a linear transformation from $F^m$ to $F^n$.

$T_3T_2T_1 = T_3(T_3^{-1}TT_1^{-1})T_1 = T$. This gives us the canonical decomposition of $T$.

Dependency for:

1. Symmetric operator iff hermitian
2. Eigenvalues and Eigenvectors

Info:

• Depth: 10
• Number of transitive dependencies: 46

Transitive dependencies:

1. /linear-algebra/vector-spaces/condition-for-subspace
2. /linear-algebra/matrices/gauss-jordan-algo
3. /sets-and-relations/composition-of-bijections-is-a-bijection
4. /sets-and-relations/equivalence-relation
5. Group
6. Ring
7. Polynomial
8. Integral Domain
9. Comparing coefficients of a polynomial with disjoint variables
10. Field
11. Vector Space
12. Linear independence
13. Span
14. Linear transformation
15. Composition of linear transformations
16. Vector space isomorphism is an equivalence relation
17. Semiring
18. Matrix
19. Stacking
20. System of linear equations
21. Product of stacked matrices
22. Matrix multiplication is associative
23. Reduced Row Echelon Form (RREF)
24. Matrices over a field form a vector space
25. Row space
26. Elementary row operation
27. Every elementary row operation has a unique inverse
28. Row equivalence of matrices
29. Row equivalent matrices have the same row space
30. RREF is unique
31. Identity matrix
32. Inverse of a matrix
33. Inverse of product
34. Elementary row operation is matrix pre-multiplication
35. Row equivalence matrix
36. Equations with row equivalent matrices have the same solution set
37. Basis of a vector space
38. Linearly independent set is not bigger than a span
39. Homogeneous linear equations with more variables than equations
40. Rank of a homogenous system of linear equations
41. Rank of a matrix
42. Basis of F^n
43. Coordinatization over a basis
44. Basis changer
45. Basis change is an isomorphic linear transformation
46. Vector spaces are isomorphic iff their dimensions are same