Mathematics Tutorial 2b & Linear Algebra II, Spring semester 2021 Midterm panic session, 30th May 2021, 17:00- Today:

Questions? •Midterm preparation •

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Linear Algebra II & Mathematics Tutorial 2bNagoya University, G30 Program

Spring 2021

Instructor: Henrik Bachmann

Homework 3: Determinants

•

Deadline: 30th May (23:55 JST), 2021

Exercise 1. (2+3+3 = 8 Points) We define the matrix

A =

0

@3 �1 0

2 0 0

�2 2 �1

1

A

and the polynomial

P (�) = det(A� �I3) ,

where I3 denotes the 3⇥ 3-identity matrix.

i) Calculate det(A).

ii) Find all solutions to P (�) = 0.

iii) For each solution � in ii) find a non-zero vector v 2 ker(A� �I3) and evaluate Av.Can you observe a relationship between v, � and A?

Exercise 2. (2+4 = 6 Points) (Geometric interpretation of the determinant)

We define the vectors v =

✓4

2

◆, u =

✓2

3

◆2 R2

.

i) Connect the endpoints of the vectors 0, v, u and v+u to get a parallelogram in R2. (Make a sketch)

ii) Show that the area of this parallelogram is given by det

✓4 2

2 3

◆, i.e. the determinant of the matrix

which has v and u as columns.

(Bonus: Show that this works in general, i.e. if you write two vectors in R2into the columns of a matrix

A 2 R2⇥2then | det(A)| gives the area of the parallelogram spanned by them.)

Exercise 3. (6+2 = 8 Points)

i) Show that the determinant is linear in each row, i.e. for any A = (ai,j) 2 Rn⇥nand 1 l n show

that the map

FA,l : Rn �! Rx 7�! det(A(l;x))

is linear. Here A(l;x) denotes the matrix A, where the l-th row is replaced by the vector xT.

(See at the bottom of page 7 in the overview notes)

ii) Assume that A is invertible. What is the kernel of FA,n?

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Linear Algebra I - Midterm preparationNagoya University, G30 Program

Spring 2021

Instructor: Henrik Bachmann

Possible midterm type exercises from the homework: HW1 Ex. 1, Ex. 2, HW2 Ex. 2.

1) Decide if the following statements are true or false. Justify your answer by giving a short explanation.

i) The set U = {f 2 C1(R,R) | f 00

= 2f} is a subspace of C1(R,R).

ii) The set U =�A 2 R2⇥2 | det(A) 6= 0

is a subspace of R2⇥2

.

iii) If (b1, b2) is a basis of a vector space V , then (b1 + b2, b2 � b1) is also a basis of V .

iv) If (f1, f2, f3) is a basis of P2, then (f 01, f

02, f

03) is also a basis of P2.

v) For any real number a 2 R there exist at least one linear map F : P1 ! P1 with det(F ) = a.

2) Calculate the determinant of the following matrix

A =

0

@1 2 �3

4 0 �1

1 �2 3

1

A .

3) We define the following elements in P2

b1(x) = x2+ 1, b2(x) = 2x� 1, b3(x) = 1 ,

and define the following function

F : P2 �! P2

p 7�! p0 + 2p .

i) Show that B = (b1, b2, b3) is a basis of P2.

ii) Show that F is a linear map.

iii) Calculate [F ]B .

iv) Determine the determinant of F .

4) Let A 2 R2⇥2and define

UA = {v 2 R2 | Av = 2v} .

i) Show that UA is a subspace of R2⇥2for any A 2 R2⇥2

.

ii) Find A,B,C 2 R2⇥2, such that dim(UA) = 0, dim(UB) = 1, and dim(UC) = 2.

5) Let V be a finitely generated vector space. Show that a linear map F : V ! V is invertible if and

only if det(F ) 6= 0.

6) Show by induction that for all n � 1 we have

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i=1

1

i(i+ 1)=

n

n+ 1.

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