Linear Independence: u,v,w Homework Ques.

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  • #1
JeeebeZ
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Homework Statement



If vectors u v & w ARE linearly independent are:

a) u+v, u+w, v+w
b) u-v, u-w, v-w

The attempt at a solution

I don't really know where to start with this. It isn't homework. It is just one of the questions out of my textbook(ch 2.3 #43 linear algebra by poole 3rd ed).

So the answers are a) yes b) no. But the question says to explain why and the back of the book just says yes and no. I don't understand why either would be different.

I guessed and said the first one would be independent. But guessing doesn't really do anything in the real world. other then gives you a small percentage to be right.

Any help would be muchly appreciated
 
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  • #2
Start with the definition. Can you or can you not find constants ##A,B,C## not all zero such that$$
A(u+v) + B(u+w) + C(v+w)=0$$
 
  • #3
LCKurtz said:
Start with the definition. Can you or can you not find constants ##A,B,C## not all zero such that$$
A(u+v) + B(u+w) + C(v+w)=0$$

$$u(A+B) + v(A+C) + w(B+C) = 0$$

$$u(A+B) + v(-A+C) + w(-B-C) = 0 $$
A=1, B=-1, C=1.
1+-1=0, -1+1=0, --1-1=0

So.. By inspection.. I can make the second one a 0 without zero's.. So that would mean. They are now linearly dependent. But how would I calculate that? I don't know how to put a proof vector into a real formula.
 
  • #4
JeeebeZ said:
$$u(A+B) + v(A+C) + w(B+C) = 0$$

$$u(A+B) + v(-A+C) + w(-B-C) = 0 $$
A=1, B=-1, C=1.
1+-1=0, -1+1=0, --1-1=0

So.. By inspection.. I can make the second one a 0 without zero's.. So that would mean. They are now linearly dependent. But how would I calculate that? I don't know how to put a proof vector into a real formula.

In the first case, you need A+B=0, A+C=0, B+C=0, because these are the coefficients of u, v and w, and u,v,w are linearly independent. The only solution is A = B = 0, so that means the original set {u+v,u+w,v+w} is linearly independent as well.

In the second case you need A+B=0, -A+C=0, -B-C=0. There is a nonzero solution, so the set {u-v,u-w,v-w} is linearly dependent.

RGV
 

What is linear independence?

Linear independence refers to a set of vectors in a vector space that cannot be written as a linear combination of each other. In other words, no vector in the set can be expressed as a sum of multiples of the other vectors in the set.

Why is linear independence important?

Linear independence is important because it allows us to determine whether a set of vectors can form a basis for a vector space. If a set of vectors is linearly independent, then it can span the entire vector space and can be used to represent any other vector in that space.

How do you determine if a set of vectors is linearly independent?

A set of vectors is linearly independent if the only solution to the equation c1v1 + c2v2 + ... + cnvn = 0, where c1, c2, ..., cn are scalars and v1, v2, ..., vn are the vectors in the set, is when all the scalars are equal to 0.

Can a set of linearly dependent vectors be used as a basis for a vector space?

No, a set of linearly dependent vectors cannot form a basis for a vector space. This is because if the vectors are linearly dependent, then some of them can be expressed as a linear combination of the others, which means they cannot span the entire vector space.

How can you determine the linear independence of a set of vectors in higher dimensions?

In higher dimensions, you can use the concept of determinants and matrices to determine the linear independence of a set of vectors. The determinant of a matrix formed by the vectors should be non-zero for the set to be linearly independent.

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