# Honours Algebra MATH10069 Assignment

Honours Algebra MATH10069

##### Friday 11th 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.

##### Attempt all questions

Importantinstructions

• Start each question on a new sheet of paper.
• 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.

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= IdVthen 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 ATare the same. (f) Let R= R[x] and I= R<x3 + 3x+ 7 >. Then

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

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

Σn1

(P, Q) = P(i)Q(i), ∀P, Q∈ R[x]<n.

i=1

1. The following defines an inner product on C2,

x2

y2

1

1x1,y1= 4xy

• 2xy
• 2xy

2

2

2

1 + 3xy,

1

x2

y2

2 where x= x1 , y= y1 ∈ C2.

1. A nonzero vector vVcannot be both in the kernel and image of a linear mapping f: VV.
2. There exist linear maps f: FnFnsuch that ker f= 0 and ker f2 /= 0.
##### [30 marks]

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

pn1xn1 + … + p0F[x] then P(A) = pnAn+ pn1An1 + … + p0 Idn.

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

1. A 0 λ 0 λ Calculate m(x) for A= λ 0 and A= λ 1 , λF. [5marks]
1. Show that IA:= {Q(x) ∈ F[x] | Q(A) = 0} is an ideal in F[x] generated by

mA(x). [5 marks]

1. Prove that mA(x) = mP1AP(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(mA1

(x), mA2

(x)),

where lcm means “theleastcommonmultiple”. [7 marks]

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

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

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

αA: Rn\ {0} → R, by αA(v) = vTAv.

1. A Let A= 2 1 . Show that Im α

−1 2

= (0, ∞). [4 marks]

1. — { } Suppose that AT= A. Prove that Im αA= 0 and that Idn+Ais invertible. [6marks]
2. Suppose that A= MTMfor some M ∈ Mat(n, R). Prove that Im αA

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

1. Suppose that AT= A. Then by the Spectral Theorem Ais diagonalisable with real eigenvalues λ1,. . . ,λn. Prove that

Im αA=

n

(Σi=1

aiλi| ai≥ 0 with (a1, … , an) /= (0, … , 0)) .

1. Show that the following are equivalent for AT= A,

(i) Im αA= (0, ∞),

1. All the eigenvalues of Aare positive,
2. A = MTMfor 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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