Honours Algebra MATH10069

Friday 11^th December 2020 1300-1500 ^{† *}

^† All students: you have an additional 1 hour to assemble and submit your PDF. Final submission deadline: 16:00.

• Students with a Schedule of Adjustment: You are entitled to a further fixed additional 1 hourfor this remote examination.

Final submission deadline: 17:00

Attempt all questions

Importantinstructions

• Start each question on a new sheet of paper.
• Number your sheets of paper to help you scan them in order.
• Only write on one side of each piece of paper.
• If you have rough work to do, simply include it within your overall answer – put brackets at the start and end of it if you want to highlight that it is rough work.

IneachofQuestions1,2and3youmayuseanearlierpartofthatquestionwhenansweringthelaterparts,evenifyouhavenotmanagedtocompletetheearlierparts.

1. For each part state whether the statement is True or False, giving a justification or counterexample for your answer. (Each part is 3 marks.)

1. Every noncommutative ring has a zero divisor.
2. → → ◦ If f: V Wand g: W Vare linear maps such that g f= Id_Vthen Vand

Whave the same dimension.

1. If f: R²ⁿ⁺¹ → R²ⁿ⁺¹ is a linear mapping then fhas an eigenvalue.
2. The group of units (Z/mZ)^×is cyclic. (Here m∈ Z_>0.)
3. Let A∈ Mat(n, F). The eigenvalues of Aand A^Tare the same. (f) Let R= R[x] and I= _R<x³ + 3x+ 7 >. Then

(x² + 1) + I(2x² + 3x) + I= (−4x² − 20x− 21) + I.

7. The following defines an inner product on R[x]_<n:= {P∈ R[x] | deg(P) <n},

Σn−1

(P, Q) = P(i)Q(i), ∀P, Q∈ R[x]_<n.

i=1

7. The following defines an inner product on C²,

x₂

y₂

1

1_{x1,y1= 4xy}

• 2xy
• 2xy

2

2

2

1 + 3xy,

1

x₂

y₂

2 where →x= ^x1 , →y= ^y1 ∈ C².

7. A nonzero vector →v∈ Vcannot be both in the kernel and image of a linear mapping f: V→ V.
8. There exist linear maps f: Fⁿ→ Fⁿsuch that ker f= 0 and ker f² /= 0.

[30 marks]

[Please turn over]

2. Let A∈ Mat(n, F), where Fis an arbitrary field. Recall that if P(x) = p_nxⁿ+

p_n−1xⁿ⁻¹ + … + p₀ ∈ F[x] then P(A) = p_nAⁿ+ p_n−1Aⁿ⁻¹ + … + p₀ Id_n.

Let m_A(x) be a nonzero monic polynomial of minimal degree such that m_A(A) = 0.

1. A 0 λ 0 λ Calculate m(x) for A= ^{λ 0} and A= ^{λ 1} , λ∈ F. [5marks]

1. Show that I_A:= {Q(x) ∈ F[x] | Q(A) = 0} is an ideal in F[x] generated by

m_A(x). [5 marks]

1. Prove that m_A(x) = m_P−1_AP(x) for any invertible P∈ Mat(n, F). [5 marks]

1. If A= ^{A1 0} is a block matrix prove that m

0 A₂ ^A

(x) = lcm(m_A1

(x), m_A2

(x)),

where lcm means “theleastcommonmultiple”. [7 marks]

1. — | Prove that m_A(λ) = 0, that is (x λ) m_A(x), if and only if λis an eigenvalue of
   1. [7 marks]
2. i=1 If p(x) = ^Qm(x− λ_i)^di (where the λ_i∈ Fare distinct) find a matrix Asuch

that m_A(x) = p(x). [6 marks]

3. (a) Let A∈ Mat(n, R) and define a mapping

α_A: Rⁿ\ {^→0} → R, by α_A(→v) = →v^TA→v.

1. A Let A= ^{2 −1} . Show that Im α

−1 2

= (0, ∞). [4 marks]

2. — { } Suppose that A^T= A. Prove that Im α_A= 0 and that Id_n+Ais invertible. [6marks]
3. Suppose that A= M^TMfor some M ∈ Mat(n, R). Prove that Im α_A⊆

[0, ∞) and that Im α_A⊆ (0, ∞) if Mis invertible. [6 marks]

4. Suppose that A^T= A. Then by the Spectral Theorem Ais diagonalisable with real eigenvalues λ₁,. . . ,λ_n. Prove that

Im α_A=

n

(_Σi=1

a_iλ_i| a_i≥ 0 with (a₁, … , a_n) /= (0, … , 0)⁾ .

5. Show that the following are equivalent for A^T= A,

(i) Im α_A= (0, ∞),

1. All the eigenvalues of Aare positive,
2. A = M^TMfor some invertible M∈ Mat(n, R).

[6 marks]

[7marks]

∈ (b) Let A Mat(n, F) be such that A² = A. Prove that Ais diagonalizable.

[6 marks]

[End of Paper]

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