Linear Algebra I & Mathematics Tutorial 1b (menti.com : 1442 2026) Tutorial 3, October 19th 2021, 14:45 - 15:29 Today:

Homework 2 •Recall Lecture 3 •Little quiz (anonymously in menti) •

While you wait

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Linear Algebra I & Mathematics Tutorial 1bNagoya University, G30 Program

Fall 2021Instructor: Henrik Bachmann

Homework 2: Matrices, vectors & the rank of a matrix

•

Deadline: 31th October, 2021

Exercise 1. (3+3 = 6 Points) Show that for all A ∈ Rm×n, x, y ∈ Rn and λ ∈ R we have

i) A(x+ y) = Ax+Ay,

ii) A(λx) = λ(Ax).

(Without using Proposition 2.4. from the lecture).

Exercise 2. (4 Points) Let p(x) = a0+a1x+a2x2+a3x3 be a polynomial of degree 3 with real coefficientsa0, a1, a2, a3 ∈ R. For this polynomial p we define the vector vp by

vp =

a0a1a2a3

∈ R4 .

Find a matrix D ∈ R4×4, such that vp′ = Dvp, where p′ denotes the derivative of the polynomial p withrespect to x. What is the rank of D?

Exercise 3. (4+3+1 = 8 Points) Let a, b, c, d ∈ R and A =

(a bc d

).

i) Show that rk(A) = 2 if and only if ad− bc #= 0.

ii) What can you say about a, b, c, d if rk(A) = 1? Consider the following subset of R2

L = {x ∈ R2 | x = Av for some v ∈ R2}.

How does L look like if rk(A) = 1? How does it look like if rk(A) = 2?

iii) What can you say about a, b, c, d if rk(A) = 0?

Version: October 15, 2021- 1 -

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Recall Lecture 3 Sets Functions

R R R N 1,23

Injective Surjective

1 f R R sur in

X 2x bijective

f RTRx E

2 f N N Not surjective

3 im fX 2x

Injective

3 f R R Not surjectivei im f

x 8Injective

4 f RI R surjective Im R

E X X2For any ae R we can

choose 8 with

f a

Not injective

3 f fl 8 but

Htl

5 f R RI HEY surjective

For any b b ERwe can solve Ax b

Injectivelater indetail

6 f RI R

E IIINot surjective

g im f

Not injective

8 f 8 f but

Htl

7 f R R Not surjective

s imfE Y injective

Surjective8 f RTR 9 ein f for

E KEY any aberbecause 9

Not injective

I AMI fl