Algebra 2 Notes 4 Flashcards
(4D-N1-4.1) nxn matrix diagonalisable
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(4P-N1-4.2) Let v1, v2, .., vn ∈ Fn and let P be the n × n matrix whose columns are v1, …, vn. Then the following are equivalent: (i) {v1, …, vn} is LI. (ii) {v1, …, vn} is a basis for F(n) (iii) P is invertible.
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(4D-N1-4.3) Eigenvalue, Eigenvector
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(4P-N1-4.4) Basic criterion for diagonalizability. Proving two things equivalent
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(4P-N2-4.5) Let A be an n×n matrix over F and λ ∈ F. Then the following are equivalent
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(4A-N2-4.6) To find a formula for Am (m ∈ N)
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(4A-N3-4.7) Solve simultaneous linear difference equations
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(4A-N3-4.8) Solve simultaneous linear differential equations
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(4D-N4-4.9) Let A be a n × n matrix over F. Then the characteristic polynomial cA(t) of A is given by cA(t) = det(tIn − A).
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(4T-N4-4.10) The Fundamental Theorem of Algebra
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(4T-N4-4.11) Let A ∈ Mn(F), and suppose that A has n distinct eigenvalues. Then A is diagonalizable.
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(4M-N5-4.12) How to diagonalise a n × n matrix with n distinct eigenvalues
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(4-N5-4.13) Definition of Subspace, Sum of two subspaces, Prop - Let U, W be subspace of V , then U + W and U ∩W are subspaces of V. Theorem (not proven!) Let U, W ≤ V . Then dim(U + W) = dim(U) + dim(W) − dim(U ∩ W)
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(4D-N5-4.17) Let U,W ≤ V. The sum U +W is direct, and write… if …
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(4D-N5-4.18) Sum of subspaces
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(4D-N5-4.19) Sum of subspaces direct
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(4L-N6-4.20) Let Ui ≤ V . Then the following are equivalent; sum Ui is direct, sum ui = 0 implies ui = 0 (probably tricky to understand)
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(4L-N6-4.21) Let Ui ≤ V , and suppose that i Ui is direct. Let Bi be a basis for Ui. Then (i) B = ∪iBi is a basis for ⊕iUi (ii) dim(⊕iUi) = i dim(Ui).
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(4D-N6-4.22) Let λ be an eigenvalue of an n × n matrix A. Then the eigenspace associated with λ is Eλ = {v ∈ Fn : Av = λv}.
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(4P-N6-4.23) Eλ is a subspace of Rn.
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(4P-N6-4.24) Let λ1,…,λr be the distinct eigenvalues of the n × n matrix A. Then the sum ri=1 Eλi is direct.
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(4D-N7-4.25) algebraic, geometric multiplicity
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(4T-N7-4.26) Let A be as in Defn 4.24. Then A is diagonalizable if and only if ei = fi for each i (i.e. the corresponding algebraic and geometric multiplicities agree). (need L 4.27)
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(4L-N7-4.27) With notation as above, ei ≤ fi
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(4M-N9-4.28) How to diagonalise a n × n matrix when possible
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(4D-N9-4.29) Two matrices similar
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(4L-N9-4.30) If A and B are similar then cA(t) = cB(t).
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(4P-N10-4.31) Let A be an n × n matrix over F. Then there exists a non-zero polynomial f(t) ∈ F[t] such that f(A) = 0.
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(4T-N10-4.32) Let A be an n × n matrix over F. Then (i) There exists a unique monic polynomial of least degree (denoted m or mA) such that m(A) = 0; this is called the minimal polynomial of A. (ii) if f is any polynomial such that f(A) = 0 then f is a multiple of m;
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(4T–) Cayley - Hamilton
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