Algebra 2 Notes 4

Question 1

(4D-N1-4.1) nxn matrix diagonalisable

Answer 1

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Question 2

(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.

Answer 2

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Question 3

(4D-N1-4.3) Eigenvalue, Eigenvector

Answer 3

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Question 4

(4P-N1-4.4) Basic criterion for diagonalizability. Proving two things equivalent

Answer 4

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Question 5

(4P-N2-4.5) Let A be an n×n matrix over F and λ ∈ F. Then the following are equivalent

Answer 5

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Question 6

(4A-N2-4.6) To find a formula for Am (m ∈ N)

Answer 6

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Question 7

(4A-N3-4.7) Solve simultaneous linear difference equations

Answer 7

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Question 8

(4A-N3-4.8) Solve simultaneous linear differential equations

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Question 9

(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).

Answer 9

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Question 10

(4T-N4-4.10) The Fundamental Theorem of Algebra

Answer 10

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Question 11

(4T-N4-4.11) Let A ∈ Mn(F), and suppose that A has n distinct eigenvalues. Then A is diagonalizable.

Answer 11

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Question 12

(4M-N5-4.12) How to diagonalise a n × n matrix with n distinct eigenvalues

Answer 12

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Question 13

(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)

Answer 13

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Question 14

(4D-N5-4.17) Let U,W ≤ V. The sum U +W is direct, and write… if …

Answer 14

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Question 15

(4D-N5-4.18) Sum of subspaces

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Question 16

(4D-N5-4.19) Sum of subspaces direct

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Question 17

(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)

Answer 17

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Question 18

(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).

Answer 18

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Question 19

(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}.

Answer 19

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Question 20

(4P-N6-4.23) Eλ is a subspace of Rn.

Answer 20

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Question 21

(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.

Answer 21

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Question 22

(4D-N7-4.25) algebraic, geometric multiplicity

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Question 23

(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)

Answer 23

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Question 24

(4L-N7-4.27) With notation as above, ei ≤ fi

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Question 25

(4M-N9-4.28) How to diagonalise a n × n matrix when possible

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Question 26

(4D-N9-4.29) Two matrices similar

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Question 27

(4L-N9-4.30) If A and B are similar then cA(t) = cB(t).

Answer 27

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Question 28

(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.

Answer 28

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Question 29

(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;

Answer 29

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Question 30

(4T–) Cayley - Hamilton

Answer 30

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