Math 2.0

Linear and multilinear algebra; matrix theory Proof

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

Builds on DeterminantDeterminant of a linear mapMatrix of a linear mapChange of basis formulaChange of basis matrixDeterminant is multiplicativeInvertible matrixDeterminant of the identity matrix

Proof

We show that every basis gives the same value of det⁡[T]B\det [T]_B. Then the definition of det⁡T\det T gives this value for each basis.

  1. Let n=dim⁡Vn = \dim V. The determinant is defined for n×nn \times n matrices with n≥1n \ge 1, so n≥1n \ge 1 here.
  2. Let BB and B′B' be two bases of VV. Let [T]B[T]_B and [T]B′[T]_{B'} be the matrices of TT in these bases.
  3. Let PP be the change of basis matrix from BB to B′B'. By the change of basis formula, PP is invertible and [T]B′=P−1[T]BP[T]_{B'} = P^{-1} [T]_B P.
  4. By the product rule, det⁡[T]B′=det⁡(P−1)det⁡[T]Bdet⁡P\det [T]_{B'} = \det(P^{-1}) \det [T]_B \det P.
  5. The ring FF is commutative, so the right side equals det⁡(P−1)det⁡P⋅det⁡[T]B\det(P^{-1}) \det P \cdot \det [T]_B.
  6. By the product rule, the definition of the inverse and the determinant of the identity, det⁡(P−1)det⁡P=det⁡(P−1P)=det⁡In=1\det(P^{-1}) \det P = \det(P^{-1} P) = \det I_n = 1.
  7. By steps 4 to 6, det⁡[T]B′=det⁡[T]B\det [T]_{B'} = \det [T]_B for any two bases BB and B′B'.
  8. The determinant of TT is defined as det⁡[T]B′\det [T]_{B'} for a chosen basis B′B'. By step 7, det⁡T=det⁡[T]B\det T = \det [T]_B for the given basis BB.

Idea

Changing the basis replaces the matrix [T]B[T]_B by a similar matrix P−1[T]BPP^{-1} [T]_B P. The determinant is multiplicative, so the factors det⁡P−1\det P^{-1} and det⁡P\det P cancel.

Reading path

To understand this page from the ground up, read these 174 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 decomposition of a permutation into transpositions, Proof that the symmetric group is generated by transpositions, Proof of the commutativity of addition by induction, Proof that equivalence classes form a partition, 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 determinant is multiplicative by expanding the rows, Proof that permuting rows multiplies the determinant by the sign, Proof that disjoint cycles commute, Proof that a matrix with two equal rows has determinant zero, Proof of strong induction from ordinary induction, Proof of the laws of exponents in a group, Proof that an intersection of subgroups is a subgroup, Proof that the determinant of the identity matrix is one, Proof that every linearly independent set extends to a basis, by Zorn's lemma, Proof of Zorn's lemma, Proof of the decomposition of a permutation into disjoint cycles, Proof of the well-ordering principle from strong induction, Proof that n independent or n spanning vectors in an n-dimensional space form a basis, Proof of the division algorithm, Proof of the Leibniz formula from the defining properties of the determinant, Proof of the uniqueness of inverses, Proof that a subspace of a finite-dimensional space has smaller or equal dimension, Proof that the determinant is multilinear in the rows, 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, Proof of the subgroup criterion.

  1. Basis concept
  2. Binary operation definition
  3. Change of basis matrix concept
  4. Coordinate space object
  5. Coordinate vector concept
  6. Cycle notation concept
  7. Definition by recursion concept
  8. Dimension concept
  9. Divisibility definition
  10. Element definition
  11. Group concept
  12. Identity matrix object
  13. Image of a linear map concept
  14. Invertible matrix concept
  15. Kernel of a linear map concept
  16. Linear independence concept
  17. Linear map concept
  18. Linear subspace concept
  19. Mathematical induction concept
  20. Matrix definition
  21. Matrix multiplication concept
  22. Matrix of a linear map concept
  23. Multilinear map concept
  24. Natural numbers object
  25. Nullity concept
  26. Peano axioms concept
  27. Permutation concept
  28. Proof of Zorn's lemma proof
  29. Proposition concept
  30. Rank of a linear map concept
  31. Spanning set concept
  32. Standard basis object
  33. Transposition concept
  34. Vector space definition
  35. Identity element definition
  36. Set definition
  37. Ring definition
  38. Subgroup definition
  39. Symmetric group definition
  40. Matrix addition definition
  41. Matrix multiplication definition
  42. Matrix–vector product definition
  43. Square matrix definition
  44. Successor function definition
  45. Conjunction definition
  46. Disjunction definition
  47. Implication definition
  48. Predicate definition
  49. Linear combination definition
  50. Linear map definition
  51. Linear subspace definition
  52. Inverse element definition
  53. Empty set object
  54. Family of sets definition
  55. Ordered pair definition
  56. Proof of the associativity of matrix multiplication proof
  57. Proof of the distributivity of matrix multiplication proof
  58. Transposition definition
  59. Proof that an intersection of subgroups is a subgroup proof
  60. Intersection definition
  61. Set-builder notation concept
  62. Union definition
  63. Invertible matrix definition
  64. Biconditional definition
  65. Addition of natural numbers definition
  66. Existential quantifier definition
  67. Universal quantifier definition
  68. Linear independence definition
  69. Endomorphism definition
  70. Image of a linear map definition
  71. Kernel of a linear map definition
  72. Linear span definition
  73. Multilinear map definition
  74. The successor function is injective statement
  75. Zero is not a successor statement
  76. Cartesian product definition
  77. Disjoint sets definition
  78. Associativity of matrix multiplication statement
  79. Proof of the uniqueness of inverses proof
  80. Distributivity of matrix multiplication statement
  81. Union of a family of sets definition
  82. Axiom of extensionality statement
  83. Subset definition
  84. Proof of the commutativity of addition by induction proof
  85. Uniqueness quantifier definition
  86. Order on the natural numbers definition
  87. Spanning set definition
  88. Alternating multilinear map definition
  89. Binary relation definition
  90. Partition of a set definition
  91. Uniqueness of inverses statement
  92. Determinant definition
  93. Finite-dimensional vector space definition
  94. Commutativity of addition statement
  95. Basis definition
  96. Proof of the Steinitz exchange lemma by induction proof
  97. Proof of strong induction from ordinary induction proof
  98. Antisymmetric relation definition
  99. Reflexive relation definition
  100. Symmetric relation definition
  101. Transitive relation definition
  102. Proof that a matrix with two equal rows has determinant zero proof
  103. Function definition
  104. Dimension definition
  105. Proof of the subgroup criterion proof
  106. Coordinate vector definition
  107. Dimension theorem for vector spaces statement
  108. Proof of the laws of exponents in a group proof
  109. Proof that the determinant of the identity matrix is one proof
  110. Steinitz exchange lemma statement
  111. Strong induction statement
  112. Proof that the determinant is multilinear in the rows proof
  113. Equivalence relation definition
  114. Preorder definition
  115. Determinant of a matrix with two equal rows statement
  116. Choice function definition
  117. Injective function definition
  118. Subgroup criterion statement
  119. Change of basis matrix definition
  120. Determinant of the identity matrix statement
  121. Determinant is multilinear in the rows statement
  122. Matrix of a linear map definition
  123. Laws of exponents in a group statement
  124. Proof that the coordinate space F^n has dimension n proof
  125. Proof that a subspace of a finite-dimensional space has smaller or equal dimension proof
  126. Proof of the well-ordering principle from strong induction proof
  127. Rank of a linear map definition
  128. Proof that n independent or n spanning vectors in an n-dimensional space form a basis proof
  129. Nullity definition
  130. Partial order definition
  131. Proof that disjoint cycles commute proof
  132. Axiom of choice statement
  133. Equivalence class definition
  134. Intersection of subgroups is a subgroup statement
  135. Dimension of the coordinate space statement
  136. Determinant of a linear map definition
  137. Linearly independent set of the right size is a basis statement
  138. Dimension of a subspace statement
  139. Well-ordering principle statement
  140. Maximal element definition
  141. Total order definition
  142. Upper bound definition
  143. Proof that equivalence classes form a partition proof
  144. Disjoint cycles commute statement
  145. Subgroup generated by a subset definition
  146. Proof of the division algorithm proof
  147. Proof that a subspace of full dimension is the whole space proof
  148. Chain definition
  149. Equivalence classes form a partition statement
  150. Division algorithm statement
  151. A subspace of full dimension is the whole space statement
  152. Proof of the decomposition of a permutation into disjoint cycles proof
  153. Zorn's lemma statement
  154. Decomposition of a permutation into disjoint cycles statement
  155. Proof that every linearly independent set extends to a basis, by Zorn's lemma proof
  156. Proof of the decomposition of a permutation into transpositions proof
  157. Every linearly independent set extends to a basis statement
  158. Decomposition of a permutation into transpositions statement
  159. Proof of the rank–nullity theorem by extending a basis of the kernel proof
  160. Sign of a permutation definition
  161. Proof that the symmetric group is generated by transpositions proof
  162. Rank–nullity theorem statement
  163. The symmetric group is generated by transpositions statement
  164. Proof that a one-sided inverse of a square matrix is two-sided proof
  165. One-sided inverse of a square matrix is two-sided statement
  166. Proof that permuting rows multiplies the determinant by the sign proof
  167. Permuting rows multiplies the determinant by the sign statement
  168. Proof of the change of basis formula proof
  169. Change of basis formula statement
  170. Proof of the Leibniz formula from the defining properties of the determinant proof
  171. Leibniz formula for determinants statement
  172. Proof that the determinant is multiplicative by expanding the rows proof
  173. Determinant is multiplicative statement
  174. Proof that the determinant 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/Determinant, Determinant of an endomorphism
About this page
TagYTKK8764G7
URIm2://YTKK8764G7
Kindproof
WritingW0 stub
MSC15A04, 15A15
ProvesDeterminant of a linear map is the determinant of its matrix
UsesDeterminant, Determinant of a linear map, Matrix of a linear map, Change of basis formula, Change of basis matrix, Determinant is multiplicative, Invertible matrix, Determinant of the identity matrix
Changed4 November 2026

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