anonymous
  • anonymous
for those who boss linear algebra.
Mathematics
  • Stacey Warren - Expert brainly.com
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katieb
  • katieb
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
anonymous
  • anonymous
1 Attachment
anonymous
  • anonymous
i actually need help with it though...
anonymous
  • anonymous
@.Sam.

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anonymous
  • anonymous
any ideas Hero?
Mendicant_Bias
  • Mendicant_Bias
I have no idea what this is, but it looks unpleasant. Good luck with that.
anonymous
  • anonymous
haha thanks. surely someone can do this :(
anonymous
  • anonymous
have you solved this kind of problems in class?
anonymous
  • anonymous
mostly, orthonormal basis, projections etc everything orthogonals..but never came across something like this.
anonymous
  • anonymous
oh okay, don't worry we can work together to solve this. Ready? :)
anonymous
  • anonymous
YES
anonymous
  • anonymous
Ok|dw:1377679515993:dw|
anonymous
  • anonymous
so, the first question what is the dimension of this base?
anonymous
  • anonymous
well if we have two elements that are different in V we get 0 however if we have two of the same elements and dot them we get ij i.e v1 . v1=1...etc |dw:1377679746655:dw|
anonymous
  • anonymous
thats what i think its saying...
anonymous
  • anonymous
and a basis is when the vectors are lin independent of each other...
anonymous
  • anonymous
then im stuck
anonymous
  • anonymous
Ok, don't worry we will solve it together just hold on a sec...
anonymous
  • anonymous
kayy :)
anonymous
  • anonymous
Alright sorry for taking long, was helping someone else. So what you said previously is correct, so the span of a vector space are set of linearly independent vectors which form the basis of the vector space, right?
anonymous
  • anonymous
correct, so would that mea we just have one linearly independent set?
anonymous
  • anonymous
since the computed values of the elements are simply numerical
anonymous
  • anonymous
we need the dim span V i.e. the number of vectors that are linearly independent in the vector space of V the key to answering the question is how the dot product relates to linearly independency, any thought on that?
anonymous
  • anonymous
well if two vectors are linearly independent then they are orthogonal (dot product is 0)
anonymous
  • anonymous
good thought
anonymous
  • anonymous
and what the question stated is that all the vectors in V are orthogonal except they are the same two vectors ( i = j ) that means the dim span V is the number of vectors in V because they are all orthogonal and the dim span V = r let me know if that is clear
anonymous
  • anonymous
ohh yea right. thats clear. so since each vector is linearly independent in V since vi . vj=0 then this means that the dimspanV=r ahhh yea now i got it
anonymous
  • anonymous
so when i=j was irrelevant?
anonymous
  • anonymous
Glad you did!
anonymous
  • anonymous
yea it would be otherwise they wouldn't be lin independent
anonymous
  • anonymous
the i = j only stated that the only time the dot product is not zero is when the two vectors involved are the same it is very relevant
anonymous
  • anonymous
oh true true
anonymous
  • anonymous
nice one
anonymous
  • anonymous
:)
anonymous
  • anonymous
can we go to the (b) part?
anonymous
  • anonymous
yea lets do it i'm keen!
anonymous
  • anonymous
so we need to know what inner product is? then we can be able to get the needed expression, any thought before we go ahead?
anonymous
  • anonymous
isn't it very similar to the dot product?
anonymous
  • anonymous
just in a vector space?
anonymous
  • anonymous
yes, that is right good! ;)
anonymous
  • anonymous
|dw:1377681129822:dw|
anonymous
  • anonymous
as you stated it will be the dot product with the vectors defined on the white board do you have any question on that?
anonymous
  • anonymous
is the first symbol vector u?
anonymous
  • anonymous
and isnt' k a scalar here?
anonymous
  • anonymous
no it is a scalar, sorry dot product result in scalar, right? My drawing in not that not as I'm not an artist.
anonymous
  • anonymous
i think i had misinterpreted the question
anonymous
  • anonymous
i can see the reason for your confusion, lol.
anonymous
  • anonymous
can you write it again. i'm getting more confused haha
anonymous
  • anonymous
each of v should be a vector and u should also be a vector let's backtrack
anonymous
  • anonymous
yes, sure.
anonymous
  • anonymous
ok, in this case where v is a vector, we cannot really use the dot product form of inner product rather, we can go the way of matrix definition
anonymous
  • anonymous
so don't bother with ur previous drawing?
anonymous
  • anonymous
|dw:1377681518747:dw|
anonymous
  • anonymous
alrighty
anonymous
  • anonymous
this is so because when a 1 X r matrix multiplies a r X 1 matrix the result should be a 1 x 1 matrix which is k1 v1 + ...... + krvr dont bother with the other one
anonymous
  • anonymous
your question helped to know that it was in error because i assumed u to be a scalar, lol. :D
anonymous
  • anonymous
haha yea ;) k i follow
anonymous
  • anonymous
so we need to find the k's we could have easily found that using the inverse function but we do not have a square matrix to work with
anonymous
  • anonymous
how can we get it into a square matrix form?
anonymous
  • anonymous
not a clue
anonymous
  • anonymous
you know that v1 itself will have its own dimension as it is a vector
anonymous
  • anonymous
correct
anonymous
  • anonymous
let's assume that the dimension is n so that v1 = (v1 1, v1 2, v1 3, ..........., v1 n)
anonymous
  • anonymous
the dimension of each vector in V in n?
anonymous
  • anonymous
is n*
anonymous
  • anonymous
in v1, we have scalars though so you mean|dw:1377682078766:dw|
anonymous
  • anonymous
|dw:1377681989326:dw|
anonymous
  • anonymous
each of the column vectors are the v's written in column vector form
anonymous
  • anonymous
let me know if you have a question on that
anonymous
  • anonymous
it is just a matrix way of writing u = k1v1 + k2v2 + k3v3 + k4v4 + ...... + krvr
anonymous
  • anonymous
so we start off like this.|dw:1377682157043:dw|
anonymous
  • anonymous
|dw:1377682236614:dw|
anonymous
  • anonymous
|dw:1377682354786:dw|
anonymous
  • anonymous
where a general vector is...
anonymous
  • anonymous
brb i gotta have dinner. you can keep going on and i'l catch up if you want. won't be too long.
anonymous
  • anonymous
thanks for your help so far
anonymous
  • anonymous
what you are writing is right however, v1, v2 must each be independently have a basis of r for the dimension of the vector space V in which they are contained to be r so we need to be able to write the equation in a form that we can easily find the k's
anonymous
  • anonymous
ok, i will just go ahead and hopefully you can join me after your dinner or you will see the solution on the whiteboard
anonymous
  • anonymous
k i'm back
anonymous
  • anonymous
yep i followed that
anonymous
  • anonymous
hold on, I'm going to attach a file where I've done all the working outs one moment...
anonymous
  • anonymous
alrighty
anonymous
  • anonymous
anonymous
  • anonymous
there you go!
anonymous
  • anonymous
it seems to be a blank page
anonymous
  • anonymous
The multiplication on the right generates matrix with k1, k2, ....., kr so that is a way of using inner product to find ki as asked in the question :)
anonymous
  • anonymous
its a document you need to save it in order to open it
anonymous
  • anonymous
Do you see it now? :)
anonymous
  • anonymous
nup still getting a blank page...
anonymous
  • anonymous
Are you sure? When you click on the attachment it leads you to a new tab, right? Then a small box appears on the left side which says for you to save the document that I sent you.
anonymous
  • anonymous
You click on save to whatever place you want to save the document into, and then the microsoft word document will automatically open.
anonymous
  • anonymous
yep still getting a blank page. maybe save it as a pdf?
anonymous
  • anonymous
If this is not working then, I may need your email address so I can send it through the email and it will work-hopefully. If you don't mind!
anonymous
  • anonymous
how bout .pdf?
anonymous
  • anonymous
anonymous
  • anonymous
yea got it!
anonymous
  • anonymous
Ok, awesome! :)
anonymous
  • anonymous
i see only 5 pages of actual writing out of the 10 pages? is that what is meant to happen?
anonymous
  • anonymous
Go through the document, and let me know if you have any questions. Also yes it is supposed to be 5 pages, my bad I added the extra 5. lol
anonymous
  • anonymous
the first whiteboard doesn't matter? cause u is a vector right?
anonymous
  • anonymous
Also I'm going to leave and help some other students, if you have any questions ask. And yes it doesn't matter I just put it there to summarise.
anonymous
  • anonymous
thanks so much!
anonymous
  • anonymous
No worries, I'm glad I could help. This stuff is really easy once you get the hang of it. Just learn the rule, practise, practise, practise, because practise makes it perfect! For now, good luck with your studies! :)
anonymous
  • anonymous
thankyou! i hope this becomes easy but the only reason i'm sticking at it because i find it so interesting
anonymous
  • anonymous
you are right, it is indeed very interesting!
anonymous
  • anonymous
:) thanks again
anonymous
  • anonymous
No worries, budd.

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