Question

Linear Algebra: Determine whether this is a vector space. Either show that the necessary properties are satisfied, or give an example that at least one of them is not.

-The set of all upper-triangular m x n matrices

Answer 1

Ah, I just don't get this stuff!

Answer 2

hehe wait a sec i am just checking :)

Answer 3

I'm pretty sure this is a vector space since it satisfies all the axioms need for a vector space.

Answer 4

it is a vector space that is forsure

Answer 5

you just have to go thru all the axioms which is a pain :)

Answer 6

umm i wonder if you can assign values to the matrix to show that all axioms are true or you just have to leave them as variables

Answer 7

umm if you show using real values it is easier

Answer 8

but i am not sure what ur prof wld want

Answer 9

It would probably have to be more formal mathematical language... he docked me when I was doing a proof with a 3x3 matrix one time.

Answer 10

All you would really need to show is that the set of upper triangular matrices is closed under addition which is obvious. The other axioms are properties of matrices, or follow directly from closure of addition.

Answer 11

well i doubt her prof wld accepy that :)

Answer 12

anyways if its closed under addition not neccesairy will it be cllosed under multiplication

Answer 13

Well, closure of addition implies that u+v is still a matrix, if u, v are upper-triangular matrices. Since addition is associative, and multiplying matrices by scalars is a property of matrices that is well-defined, all the axioms are satisfied.

In a vector space, you never multiply vectors together.

Answer 14

kk but she still has to show that all the axioms are satisfied

Answer 15

Yes, but their proof are almost trivial once you've showed closure of addition.

Answer 16

True :)

Answer 17

umm brinethy u know how to show that all the 10 axioms r true?

Answer 18

I don't know how to do this with a general mxn matrix

Answer 19

showing that it's closed under scalar multiplication and vector addition, I mean.

Answer 20

give me a sec and i will do it

Answer 21

Scalar multiplication should already be given to you as a property of matrices. As for vector addition, choose two arbitrary m x n upper triangular matrices, and add them together. It should become pretty obvious that it's closed after that.

Answer 22

Pippa, thank you very much for taking the time to do this for me. I can't tell you how much I appreciate it.

Answer 23

well i just proved that they are closed under addition and scaler multiplication. Do u see how?

Answer 24

Aren't those all lower triangular matrices? not upper?

Answer 25

oh Sh* but it wld be the same thing

Answer 26

its a good thing u r around king

Answer 27

no problem at all.

Answer 28

umm there are still 8 more axioms that need to be proved

Answer 29

some of them take much longer to prive than others

Answer 30

These are the two main ones

Answer 31

R u still there?

Answer 32

Yeah, I stepped out. My dad had to tell me something

Answer 33

I'm opening your file right now

Answer 34

lol that is ok:)

Answer 35

So as long as it's closed under vector addition and scalar multiplication, the set is a vector space?

Answer 36

umm well u kind of have to prove the other axioms as well which is harder

Answer 37

and btw by accident i proved using lower traingles in stead of upper

Answer 38

Yes, I am aware of that

Answer 39

While you have to show a little more, the only other part that doesn't follow directly from those facts is that you need to find an inverse matrix (hint: multiply by -1 to get the inverse)

Answer 40

y wld u need to do that?

Answer 41

I am just wondering cuz i dont remember ever doing that?

Answer 42

idk I just went thru every axiom

Answer 43

Sorry guys, I am even more lost. I guess you have to go through all the axioms but I'm not getting a clear answer. I'll ask the instructor tomorrow, but he'll probably confuse me even more lol

Answer 44

LOL yes well you just went thru two axioms you just proved that it is closed under addition and that it is closed under scaler mutiplication

Answer 45

Like in ur textbook there shld be another 8 axioms that need to be proved liek one of them is u+v = v+u and (u+v)+w=u+(w+v) and many others

Answer 46

but it will take me hours to write them out unfortunately i dont have a tablet so like maybe ask ur prof to show u how

Answer 47

k gonna go study i hope i was a lil bit helpful :)

Answer 48

I will. And I really appreciate your help, thank you for taking the time to explain this stuff to me.

Answer 49

U r welcome its fun to teach others cuz that means I am comfortable enough in this area to explain it to others :) Gluck