Re: Linearly independent vectors/mystery behind it

On Feb 28, 8:23 pm, David C. Ullrich <dullr...@xxxxxxxxxxx> wrote:
On Sat, 28 Feb 2009 00:34:21 -0800 (PST), Nimo <azeez...@xxxxxxxxx>
wrote:

Hi,

a set of vectors { u1, u2, u3......un }

is said to be linearly 'dependent'   iff

    sigma i=1_n Ci ui = 0____(1)

where not all Ci 's = 0

Yes.

_______________________

a set of vectors { u1, u2, u3......un }

is said to be linearly 'independent'  if

     sigma i=1_n Ci ui = 0_____(2)

where all Ci 's = 0

No. _Any_ set of vectors has this property!

You need to read much more carefully.
What does the definition _really_ say?

Possibly you need a book in your native
language. We've noticed your English is
very weak. Nothing wrong with that,
but here you're going to have to understand
_exactly_ what all the little words mean,
and that may not be possible for you in
English. Just a suggestion.

In fact those vectors are independent
if the _only_ solution to equation (2)
is to take all the C_i = 0.

"the only solution to (2) is to take
all the C_i = 0"

and

"(2) holds where all the C_i = 0"

mean _totally_ different things.
But they may look the same to you in
English, hence my suggestion.

__________________________

Statement: - now, a vector U can be written as sum of

linearly 'independent' vectors

  U  = sigma i=1_n Ci ui_____(3)

my doubt is,

In eq(3) the R.H.S
is very much equal to 'zero vector'
according to the definition eq(2)

No. Definition (2) does not say that
any time you see "C_i" for the rest
of your life C_i = 0. And it does not
say that the sum = 0 for every choice
of C_i, just the opposite.

but, how can it is justified for the above " Statement "

or am I wrong anywhere..?

Well of course you are! If you're not wrong then
all of linear algebra is totally wrong - how likely
is that?

please write in brief,
it would be very helpful for me

Thank you
  so long
     nimo

David C. Ullrich

"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)

now the things are VERY clear,

slight misunderstanding,
the definition by me(in hurry); sorry for that.

up to my analysis till now,
let's see AGAIN,


A set { x1,x2,x3,......xn} of r vectors is said to be
linearly 'dependent' if there exists "r" scalars
k1,k2,k3......kr; not all zero

such that k1x1 + k2x2 + k3x3+.....krxr = 0

where 0 denotes the n vector with components all zero.
_________

A set { x1,x2,x3,......xn} of r vectors is said to be
linearly 'independent' set, if it is not linearly "dependent"

i.e. k1x1 + k2x2 + k3x3+.....krxr =! 0

[ forcibly if I've to make it like the below ]

k1x1 + k2x2 + k3x3+.....krxr = 0;

where 0 denotes the n vector with components all zero.

the only choice I'm left out is

==> k1 =0, k2 =0, k3 = 0,.......kr =0.


Am I right now ?

Thank you at all for the help
so long
nimo
.

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