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Linear Algebra II (MATH1049): Coursework Sheet 8
2024/25
Submit a single pdf with scans of your work to Blackboard by Tuesday, 6 May 2025, 17:00.
Exercise 1
From Calculus we Rknow that for any polynomial function f : R → R of degree at most n, the function
I( f ) : R → R, s 7→ 0s f (u) du, is a polynomial function of degree at most n + 1. Show that the map
I : Pn → Pn+1 ,
f 7→ I( f ),
is an injective linear transformation, determine a basis of the image of I and find the matrix M ∈
M(n+2)×(n+1) (R) that represents I with respect to the basis 1,t, . . . ,t n of Pn and the basis 1,t, . . . ,t n+1
of Pn+1 .
Exercise 2
1−i
α
i
(a) Let α ∈ C and A := i − α 1 − α α − i ∈ M3×3 (C). Compute det(A) ∈ C.
1−α
1
2+α
(b) Let F be a field, n be even and let c1 , . . . , cn ∈ F. Follow the blueprint of the proof of Example
5.2(d) and use Exercise 2(a) on Coursework Sheet 3 to compute the determinant of the matrix
0 . . . 0 c1
..
.
·
· 0
B :=
.. ∈ Mn×n (F).
0 ·
·
.
cn 0 . . . 0
Exercise 3
Let F be a field, let n ≥ 1 and let a, b ∈ F. Furthermore let
b a a ... a
a b a ... a
A := a a b . . . a ∈ Mn×n (F).
.. .. ..
..
. . .
.
a a a ... b
Show that det(A) = (b + (n − 1)a)(b − a)n−1 . (Hint: Begin with R1 7→ R1 + R2 + . . . + Rn , apply Theorem
5.3(b) and use further row operations to arrive at an upper triangular matrix.)
Exercise 4
1+i 1−i
2
2
3+i
2i
i
−i ∈ M3×3 (C) and B = 1
2 − i 2 + 2i ∈ M3×3 (C).
Let A = 3
1
2−i 2+i
1−i
i
3
Compute det(A), det(B), det(AB) and det(A3 ).
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