Let u and v be two linearly independent vectors. Let w = u + v and x = u - v. Prove that w and x are linearly independent.
i get to 0 = u (a+b) + v (a-b) and then i dont know what to do next or if this is even right
#prove it's linearly independent
pseudo copper
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Let u and v be two linearly independent vectors. Let w = u + v and x = u - v. Pr...
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uncut spade
If u and v are linearly independent, what do a+b and a-b have to equal here?
pseudo copper
is it a=b=0 so a-b=0 and a+b=0?
uncut spade
Yep
What happens when you add the two equations together
Add $a-b = 0$ to $a + b = 0$
past horizonBOT
MicMac
pseudo copper
its zero
uncut spade
On the right it's zero and on the left its (a-b) + (a+b), i.e., $(a - b) + (a + b) = 0$
past horizonBOT
MicMac
uncut spade
What does that simplify to?
pseudo copper
2a?
past horizonBOT
MicMac
pseudo copper
so i can just get rid of the u and v?
uncut spade
Not exactly. Your goal should be to show that if $a(u + v) + b(u - v) = 0$, then $a = 0$ and $b = 0$.
past horizonBOT
MicMac
uncut spade
At this point, you're halfway there (more than that really) because we just saw that a must equal 0
Now you just have to show that b = 0, which will be way easier than everything else so far.
But if you mean that we don't have to do anything else with u and v in this problem, you're totally right
All you gotta do is show that both a and b are zero
pseudo copper
is it impossible for u or v to be 0
uncut spade
Yeah, they can't be zero because the 0-vector is linearly dependent with everything
Since u and v are linearly independent they both gotta be nonzero
Like if $u$ were the zero vector, then $cu$ would equal zero for any real number c. But linear independence requires that to happen only when c is zero.
past horizonBOT
MicMac
pseudo copper
ohh i see
but i dont understand how we take 0 = u(a+b) + v(a-b) and turn it into 0 = (a+b) + (a-b)
uncut spade
So, you said back at the beginning that since u and v are linearly independent, then the numbers multiplying them here, a+b and a-b, have to be 0
So you made two equations $a+b = 0$ and $a-b = 0$
past horizonBOT
MicMac
uncut spade
I suggested that you add the two equations together to eliminate b
pseudo copper
ohhh sorry i get it now
uncut spade
It's all good
pseudo copper
thank u so much micmac
uncut spade
Sure thing! Take care