Question

Linear Algebra Proof.
Picture Attached

Answer 1

Hey man!

Answer 2

hey long time bro : ), what does R^m mean

Answer 3

I don't even do math this semester except I am mentoring students for 1st yr maths and i'm struggling with proofs haha - i don't even need linear algebra for engineering haha

i think its to do with an mxn matrix so it spans R^m?

Answer 4

oohh okay i see

Answer 5

there are M independant rows since its invertible so

Answer 6

as well as N independant columns

Answer 7

so a column must contain m, components of a vector

Answer 8

hence it spans R^m?

Answer 9

yeah

Answer 10

is that what you would write as a show that question? haha

Answer 11

i dunno wha to say lol, maybe just write |A| =/= 0 there fore M independant rows and N independant columns

Answer 12

b) non independant rows and columns there for there must exist atleast one row such that Row n=k(Rown1)+K2(Row2), therefore the matrix can be rewritten as m-1 rows atleast and result in same solution therefore must span less or equal to R^m-1 space with the columns

Answer 13

@Kainui

Answer 14

Dude hes on the move atm ....

Answer 15

so that means colA cannot span R^m because it has always has m-1 independent vectors?

Answer 16

yeah

Answer 17

wait when they say col(A) we are talking abou the column picture by taking all the colums into account right like they are all this vectors in m dimensional space

Answer 18

col A is like when you row reduce it and the pivots entries are the corresponding vecotrs that form a a column basis

Answer 19

So what do you think?

Answer 20

not sure lol lets wait for kainui

Answer 21

im not a big fan of this proofy stuff -.- I feel like its an obvious thing but i dont know all the details u have to put in

Answer 22

Mmm it is a thing like that. Its just putting the pen on the paper to write a succinct proof aye. like the knowledge is there...haha

Answer 23

ask some more linear algebra questions !! :)

Answer 24

i am starting to play with matrices more i like it

Answer 25

i have another proof haha

Answer 26

okay post that too

Answer 27

it seems straightforward

Answer 28

but duuno how to write it like a proof..kk

Answer 29

identity is an orthonormal basis right

Answer 30

yep!

Answer 31

can u write a general form for orthonormal basis that is not identity

Answer 32

ummmm

Answer 33

isn't it a basis whose vectors are orthonormal?

Answer 34

not sure

Answer 35

http://prntscr.com/4f8hts

Answer 36

yep thats in my notes!

Answer 37

A={v1,v2,v3....vn} , if A has n vectors then each vector must have n components and
vi dot vj=1, when i =j and vi dot vj=0, when i=/=j, where i and j can be and ideces between 1 and n

Answer 38

=/= means does not equal to right?

Answer 39

yah

Answer 40

and that is pretty much the definition right

Answer 41

yeah

Answer 42

now
if A^t=A-1 then
A^t*A=identity
i think we need to show that multiplying by the transform of A, where A is orthonormal will result in Ai dot Ai along the diagonals

Answer 43

you can think of
A^t={v1,
v2,
.
.
vn}

A={v1,v2,v3...Vn}

Answer 44

okay

Answer 45

only the diagonal has to exist in this case as Vi.Vj=0 where i=/=j

Answer 46

and as they are orthonomal basis the diagonal indices must go to 1

Answer 47

and thats why it equals the idenitiy matrix?

Answer 48

yeah, bt lemme see lol i feel like we are skipping steps

Answer 49

we can now say that true for a square matrix?

Answer 50

mmmm

Answer 51

yeah we did its Vn ectors so, and i said that any V must have n components if this matrix is composes of N vectors

Answer 52

this only works for square matrices i an just draw a picture in my mind maybe u can explain better