Today: Who wants to be a Linear Algebraist Rules:

There will be up to 10 questions (if we have enough time) •Each question has 4 possible answers (A,B,C,D) of which •multiple answers can be correct. Write the solution to each question in the private(!!) chat to •me like this: “Q5: A, B”, if you think A and B is correct for Question 5. You might need to do small calculations. So be prepared to write down something. •This is just for fun and it will not influence your grade. I want to get an honest overview of •the current knowledge. No need to cheat!

Linear Algebra IForiatthInne 20

QuestionTest question What is 2 2

5 B

0

4 5 1 2 2

Question I Let V be a vector spaceUN EV X µ ER

Which of thefollowing things are defined

u v

ütut µ

Correct A D

A U Uttiv EV

B The product of twovectors nu is

not defined

C du vector µ numbervector number is not defined

D d e v is definedscalarmaltvector

Feet

Question Let V be a vector space andU WC V subspaces of V

Which statements are true

tabspaceBOunwisasubst

Correct B C

A Not a subspace ExK RA span91913

W span9lbD Hd au W

B C are subspaces Homework

D Is not a subspace because 0 Vllt

Question LetV be a vector space andviii Ne E V are linearly independent

Which statements are true

In.ve is a Türen 4 14pmbasis of as a basis

ÄÄt ämwcorrect B C A This is just true if dimmt

B The space U span94 ve has

basis Iv Ne

If I 4 4 44 41 0

X V Ikt Kit X VEO

X X 0 because 4 Nt are

1in indep

D It could be that dimutldimm l is correct

Question 4 Let P XII pzkt zxi x.snB X 4 P 3

and U span Pz B C PzWhich statements are true

d dink 3

Py EU ODPEspanSR.R.B.RU

P XII RKI 2 4 2 2 p p 2 42 1ktß X 4 Pa 3 2 4 2

We see that PE 2 p 1ps but p and ßare 1in indep dim A 2

P f U because 3 7 P e Kp k7,1 411 11 41

424441 447

has no Solutionsfor X HERThis shows that puh Karelin indep zueis

Questions let p Rtl pzlxt X t.pkSet B P Pz Ps and 141 2 4 1Which statements are true

Bi G B 7

B7B

R B and ß are 1in indep B isa

basis

Pakt 2 4 1 2 p.CH I KH t OpsK

curlBkk O p to P t.pk

BIß 9

Question Let F Ps IR be a linearmap

Which statements are true

ÄÄt

demPs dim IR 4

Therefore F surjectiveF injectiveF bijectiveisomorphismimE RtKerlF SO

and dim 903 0

Theorem 2 5

Question 7 Let F Pi RzKim

B d 9 Bz HkWhich statements are true

EIB TI EIBE H

i EÄ

Bill BELE

D Flott 9

FIA f C2

B F E B E Http

5 2 C tl.czEdt L

QuestionµO I

What is the determinant of A

deHAI 3 detCAF I

det Hk O deHA

Using the formula for the det of a 3 3Matrix we get

def A I t.E ti 2 tt H I 1

2 2 1 5

Question let µ t with abersuch that def 4

ab c

Which statements are true

auf f z BOdetf.EEdet t DOdetlb.EE

The matrix la is obtained from A

by changing Rowland 2sign of detchanges

and then multiplying the first now bytzRow operations

det la 2 RzanaR3

bag is obtained from A byadding the first now to theSecond Row op Rldeterminant is still 4

Question Let A BE IR and KERWhich statements are true

AOdetlABtdeHAIdetlBUBOdeHAtBtdgetAdetlBl

detlX.A t.deHAT detHAI I.deHA

A see Lecture 7 Ahm 4.12

B detl ttd.it Fdetlhi detHi

det D 01 1 2

C Wrong if n I

D HA ii Adetl f X detHAI Idet.CH