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Flashcards

A0

Ben Tidswell Part A uni English 26 cards

A0

Cards in this deck (26)

  1. 1Define a dual basisDefine a dual basis
  2. 2Show that the dual linear map matrix is transpose to the original oneShow that the dual linear map matrix is transpose to the original one
  3. 3Define the annihilatorDefine the annihilator
  4. 4Prove that dim(U^0)=dim(V)-dim(U)Prove that dim(U^0)=dim(V)-dim(U)
  5. 5Prove that lambda is a root of m_T(x) iff lambda is an eigenvalue of TProve that lambda is a root of m_T(x) iff lambda is an eigenvalue of T
  6. 6Prove that m_T(x) has distinct roots iff T is diagonalisableProve that m_T(x) has distinct roots iff T is diagonalisable
  7. 7Prove that if T*, the adjoint of T, exists it is uniqueProve that if T*, the adjoint of T, exists it is unique
  8. 8Prove that if T:V -> V is self-adjoint and V is finite dimensional then there exists an orthonormal basis of eigenvectors for T, and hence T can be represented by a real diagonal matrixProve that if T:V -> V is self-adjoint and V is finite dimensional then there exists an orthonormal basis of eigenvectors for T, and h...
  9. 9Define the dual (dual space) of VDefine the dual (dual space) of V
  10. 10Define the dual map of a linear map T:V -> WDefine the dual map of a linear map T:V -> W
  11. 11What are the conditions for T:U -> V to be a well-defined linear mapWhat are the conditions for T:U -> V to be a well-defined linear map
  12. 12Prove that for all A in M_n(F), there exists a non-zero polynomial f(x) in F[x] s.t. f(A)=0Prove that for all A in M_n(F), there exists a non-zero polynomial f(x) in F[x] s.t. f(A)=0
  13. 13Prove that if f(A)=0 then m_A|f. Futhermore, prove that m_A is unique and hence well-definedProve that if f(A)=0 then m_A|f. Futhermore, prove that m_A is unique and hence well-defined
  14. 14State a criterion for diagonalisability in terms of m_T(x)State a criterion for diagonalisability in terms of m_T(x)
  15. 15Prove that the dual basis {e'_1,...,e'_n} is a basis for V'. In paticular the assignment e_i -> e'_i defines an isomorphism of vector spaces. In paticular, dimV = dimV'Prove that the dual basis {e'_1,...,e'_n} is a basis for V'. In paticular the assignment e_i -> e'_i defines an isomorphism of vector ...
  16. 16State a relationship between _B[T]_B and _B'[T']_B' & χ_T(x) and χ_T'(x) & m_T(x) and m_T'(x)State a relationship between _B[T]_B and _B'[T']_B' & χ_T(x) and χ_T'(x) & m_T(x) and m_T'(x)
  17. 17Prove that under the natural map V -> V'' given by v -> E_v, U is mapped isomorphically to U^(00)Prove that under the natural map V -> V'' given by v -> E_v, U is mapped isomorphically to U^(00)
  18. 18Define the minimal polynomial of T: V -> VDefine the minimal polynomial of T: V -> V
  19. 19Define the characteristic polynomial of T: V -> VDefine the characteristic polynomial of T: V -> V
  20. 20State and prove Cayley-HamiltonState and prove Cayley-Hamilton
  21. 21Define the adjoint T* of TDefine the adjoint T* of T
  22. 22Define a complex inner productDefine a complex inner product
  23. 23State the Division algorithm for polynomialsState the Division algorithm for polynomials
  24. 24State the Fundamental Theorem of AlgebraState the Fundamental Theorem of Algebra
  25. 25State the Rank-Nullity TheoremState the Rank-Nullity Theorem
  26. 26State and prove the Primary Decomposition TheoremState and prove the Primary Decomposition Theorem

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Description
uni Part A Flashcards on A0, created by Ben Tidswell on 02/01/2022.
Created by
Ben Tidswell · Jan 2022
Updated
Jun 8, 2022
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