Math 2.0

Linear and multilinear algebra; matrix theory Proof

Proof that the trace of a linear map does not depend on the basis

Builds on Trace of a linear mapTrace of a matrixMatrix of a linear mapChange of basis matrixChange of basis formulaTrace of a product is symmetricAssociativity of matrix multiplicationInvertible matrix

Proof

The trace of TT is defined as tr⁡[T]B0\tr [T]_{B_0} for some basis B0B_0 of VV. We show that every basis gives the same number.

  1. Let BB and B′B' be two bases of VV. Let M=[T]BM = [T]_B and M′=[T]B′M' = [T]_{B'} be the matrices of TT, and let n=dim⁡Vn = \dim V.
  2. Let PP be the change of basis matrix from BB to B′B'. By the change of basis formula, PP is invertible and M′=P−1MPM' = P^{-1} M P.
  3. By associativity, M′=P−1(MP)M' = P^{-1} (MP).
  4. By the symmetry of the trace of a product, applied to the n×nn \times n matrices P−1P^{-1} and MPMP, we get tr⁡(P−1(MP))=tr⁡((MP)P−1)\tr\left(P^{-1}(MP)\right) = \tr\left((MP)P^{-1}\right).
  5. By associativity again, (MP)P−1=M(PP−1)=MIn=M(MP)P^{-1} = M(PP^{-1}) = M I_n = M.
  6. By steps 3 to 5, tr⁡M′=tr⁡M\tr M' = \tr M, with the trace of a matrix. So tr⁡[T]B′=tr⁡[T]B\tr [T]_{B'} = \tr [T]_B for any two bases BB and B′B'.
  7. In particular, taking B′=B0B' = B_0, we get tr⁡T=tr⁡[T]B0=tr⁡[T]B\tr T = \tr [T]_{B_0} = \tr [T]_B for the given basis BB.

Idea

Changing the basis replaces the matrix MM by P−1MPP^{-1} M P. The trace does not see the order of the factors in a product, so the factors P−1P^{-1} and PP cancel inside the trace. This makes the trace a property of the map and not of the chosen basis.

Reading path

To understand this page from the ground up, read these 102 pages in order. The foundations come first. The path takes the proofs Proof of the Steinitz exchange lemma by induction, Proof of the associativity of matrix multiplication, Proof that a subspace of full dimension is the whole space, Proof of the change of basis formula, Proof of the distributivity of matrix multiplication, Proof that the coordinate space F^n has dimension n, Proof that the trace of a product is symmetric, Proof that every linearly independent set extends to a basis, by Zorn's lemma, Proof of Zorn's lemma, Proof that n independent or n spanning vectors in an n-dimensional space form a basis, Proof that a subspace of a finite-dimensional space has smaller or equal dimension, Proof that a one-sided inverse of a square matrix is two-sided, Proof of the rank–nullity theorem by extending a basis of the kernel.

  1. Basis concept
  2. Change of basis matrix concept
  3. Coordinate space object
  4. Coordinate vector concept
  5. Dimension concept
  6. Element definition
  7. Group concept
  8. Identity matrix object
  9. Image of a linear map concept
  10. Invertible matrix concept
  11. Kernel of a linear map concept
  12. Linear independence concept
  13. Linear map concept
  14. Linear subspace concept
  15. Matrix definition
  16. Matrix multiplication concept
  17. Matrix of a linear map concept
  18. Nullity concept
  19. Proof of Zorn's lemma proof
  20. Proposition concept
  21. Rank of a linear map concept
  22. Spanning set concept
  23. Standard basis object
  24. Trace of a linear map concept
  25. Trace of a matrix concept
  26. Vector space definition
  27. Set definition
  28. Ring definition
  29. Matrix addition definition
  30. Matrix multiplication definition
  31. Matrix–vector product definition
  32. Square matrix definition
  33. Disjunction definition
  34. Implication definition
  35. Predicate definition
  36. Linear combination definition
  37. Linear map definition
  38. Linear subspace definition
  39. Family of sets definition
  40. Ordered pair definition
  41. Proof of the associativity of matrix multiplication proof
  42. Proof of the distributivity of matrix multiplication proof
  43. Union definition
  44. Invertible matrix definition
  45. Trace of a matrix definition
  46. Existential quantifier definition
  47. Universal quantifier definition
  48. Linear independence definition
  49. Image of a linear map definition
  50. Kernel of a linear map definition
  51. Linear span definition
  52. Cartesian product definition
  53. Associativity of matrix multiplication statement
  54. Distributivity of matrix multiplication statement
  55. Union of a family of sets definition
  56. Proof that the trace of a product is symmetric proof
  57. Subset definition
  58. Uniqueness quantifier definition
  59. Spanning set definition
  60. Binary relation definition
  61. Trace of a product is symmetric statement
  62. Finite-dimensional vector space definition
  63. Basis definition
  64. Proof of the Steinitz exchange lemma by induction proof
  65. Antisymmetric relation definition
  66. Reflexive relation definition
  67. Transitive relation definition
  68. Dimension definition
  69. Coordinate vector definition
  70. Dimension theorem for vector spaces statement
  71. Function definition
  72. Steinitz exchange lemma statement
  73. Preorder definition
  74. Change of basis matrix definition
  75. Choice function definition
  76. Matrix of a linear map definition
  77. Rank of a linear map definition
  78. Proof that a subspace of a finite-dimensional space has smaller or equal dimension proof
  79. Proof that the coordinate space F^n has dimension n proof
  80. Nullity definition
  81. Proof that n independent or n spanning vectors in an n-dimensional space form a basis proof
  82. Partial order definition
  83. Dimension of the coordinate space statement
  84. Axiom of choice statement
  85. Linearly independent set of the right size is a basis statement
  86. Dimension of a subspace statement
  87. Maximal element definition
  88. Total order definition
  89. Upper bound definition
  90. Proof that a subspace of full dimension is the whole space proof
  91. Chain definition
  92. A subspace of full dimension is the whole space statement
  93. Zorn's lemma statement
  94. Proof that every linearly independent set extends to a basis, by Zorn's lemma proof
  95. Every linearly independent set extends to a basis statement
  96. Proof of the rank–nullity theorem by extending a basis of the kernel proof
  97. Rank–nullity theorem statement
  98. Proof that a one-sided inverse of a square matrix is two-sided proof
  99. One-sided inverse of a square matrix is two-sided statement
  100. Proof of the change of basis formula proof
  101. Change of basis formula statement
  102. Proof that the trace of a linear map does not depend on the basis proof

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Trace_(linear_algebra), Trace of a linear operator
About this page
TagGGAHVSTQ54
URIm2://GGAHVSTQ54
Kindproof
WritingW0 stub
MSC15A04, 15A15
ProvesTrace of a linear map is the trace of its matrix
UsesTrace of a linear map, Trace of a matrix, Matrix of a linear map, Change of basis matrix, Change of basis formula, Trace of a product is symmetric, Associativity of matrix multiplication, Invertible matrix
Changed4 November 2026

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