Question

Show that this subset of R^2 {(a,c),(b,d)} is linearly independent if and only iff ad-bc =/=0

Answer 1

This takes time, hang on...

Answer 2

alright np i have been struggling with this one for awhile

Answer 3

I can prove one way, still working on the other :)

Answer 4

ok i actually have a step by step key as well but the answers do not make sense to me

Answer 5

Oh, so you already have the answers? >.> I'll work on it anyway, what a coincidence that I'm perusing through some of these at the moment ;)

Answer 6

ya i know the steps they say to make but it seems to jump. it says to do a case wehre a=/=0 and do row reduction then you have to do it with a=0 ,c=0 and when c=/= 0

Answer 7

Ok, I can prove it now, are you interested? :)

Answer 8

yes please

Answer 9

Ok, here we go :)
{(a,c), (b,d)} is in R^2, so a,b,c,d are all in R (just to clarify)
Suppose ad - bc = 0
Then there exist scalars that we can multiply to (a,c) and (b,d), these scalars are
d(a, c) and -c(b, d), and taking the sum of these
(ad, cd) + (-bc, -cd) = (ad - bc, cd - cd) = (0, 0)

Meaning the set is linearly dependent.

So if ad - bc = 0, then the set if linearly dependent, therefore
If the set is linearly independent, ad - bc =/= 0

Get it? I'll stop here, I'll continue when you tell me to :)

Answer 10

By the way, this has the condition that not both d and c are 0, but I'll show that later ;)

Any questions so far? May I proceed?

Answer 11

yes please

Answer 12

Did you understand the first part, though?

Answer 13

ya. but why did you use -c?

Answer 14

oh never mind i see

Answer 15

OK. I'll proceed then :)

Answer 16

Suppose this time that the set is linearly dependent.
Then one of them is a linear combination of the other elements.
WLOG, suppose that
(a, c) = k(b, d) for some scalar k.

Then
(a, c) = (kb, kd)
a = kb
c = kd

Therefore

a/b = k
c/d = k
a/b = c/d
ad = bc
ad - bc = 0

Therefore, if the set is linearly dependent, then ad - bc = 0
Thus, if
ad - bc =/= 0
then the set is linearly independent

QED (end of proof)
Any questions? :)

Answer 17

haha nope tyvm
I really appreciate it

Answer 18

No problem. I'll show that if d and c are both 0, then the set is linearly dependent:
Suppose d = c = 0
Then the set is
{(a,0), (b,0)}
But it's linearly dependent, since there is a scalar, -b/a such that
(-b/a)(a, 0) + (b, 0)
(-b, 0) + (b, 0)
= (0, 0)

:)

Answer 19

:)