ByKS00001
I know what linearly independent and independent means with respect to vectors, but I don't understand how to recognize it.
a(1/0/2) and b(2/0/4)
or
2.Example:
c(1/0/0) and d(0/5/0)
How can I recognize it?
(1 votes)
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Two vectors are linearly dependent on each other if you can form one vector by multiplying another vector (you also call collinear).
Example a(1/0/2) * 2 = a(1*2/0*2/2*2) = b(2/0/4)
So if you take vector a * 2 you get vector b.
There is no common factor for the vectors c and d, i.e. they are linearly independent.
That means that when I get to the other by multiplying a vector, are they linearly dependent?
exactly
Yes
Determinant of the Matix from the three vectors would have to be 0.
please
All right, thank you
Exactly, so in this case:
—> Bsp: a=(1/0/0) b=(0/1/0) c=(1/1/0)
—> c = 1 * a + 1 * b
In this case, the factor of a and b is exactly 1. As I have already written above, it could also be another factor
Wouldn’t that be an addition? Or would it mean that you do a and b once each time?
In this case, you could form the vector c by adding a and b, so:
c = a + b
If I really understand that, for example, this might look like this:
Vectors a, b, c
c = 2 * a + 3 * b
b=(0/1/0) c=(1/1/0)
Thank you. What would it look like with, for example, 3 vectors?
Well, if you have 2 vectors, they are linearly dependent when they are colinear or multiple ones. 3 vectors are linearly dependent when they lie in one plane or are also called komplanar. So if you want to find out if 2 Vekoren are linearly dependent you see if vector a = vector B * r is! If you want to see if 3 vectors are compplanar then you set up a linear GS and solve it. If you can solve it linearly depending when it has infinitely many solutions independent. It’s really easy.
That’s probably not true, otherwise you just have to look,
whether the conditions are fulfilled.
They are linearly dependent when the one vector
by one factor from the other.
This is the case in the first two (factor=2).
The last two are not, so they are
not linearly dependent.
and what would it look like with 3 vectors?
That’s right. If the vectors A and B are linearly dependent,
See if C shows a factor from A or B.
It then also emerges from the other vector.