Express one vector as a linear combination of others

In summary, to show that the set of vectors {(1,2,3), (3,2,1), (5,5,5)} is linearly dependent, we can express one vector as a linear combination of the others by setting up a system of equations and solving for the parameters a, b, and c. After eliminating c and b, we find that a is an arbitrary parameter. Choosing a value for a and plugging it back into the original equation, we can show that the set is indeed linearly dependent.
  • #1
Saladsamurai
3,020
7

Homework Statement



Show that the set of vectors is linearly dependent (LD) by expressing one vector as a linear combination (LC) of the others:

{(1,2,3), (3,2,1), (5,5,5)}


The Attempt at a Solution



I would like to do this systematically (without guess and check). So I assumed that if the set is LD, then there exists some values of a,b,c not all zero such that:

a(1,2,3) + b(3,2,1) + c(5,5,5) = 0

hence,

a + 3b +5c = 0
2a + 2b +5c = 0
3a + 1b +5c = 0

Now I should be able to solve for 2 of the parameters (assuming only one of the equations is not independent, else I an solve for 1 ...)) a,b,c in terms of the third which would remain arbitrary.

Eliminating c from the first and second equations, we find that a = b .

Eliminating b, from the second and third equations we find that c = -4a/5 .

Now, if I plug these values back into anyone of the equations, I simply get the identity. What am I missing here? Am I messing something up? Or are they ALL multiples of each other?

Thanks!
 
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  • #2
I'm confused. Haven't you done what you wanted to? You seem to have found everything in terms of a.
 
  • #3
hgfalling said:
I'm confused. Haven't you done what you wanted to? You seem to have found everything in terms of a.

Oh. Maybe I have :redface: So here, a is my arbitrary parameter. Say, I let a = 1, then b = 1 and c = -4/5. Plugging these into the original equation, yields:

(1,2,3) + (3,2,1) - (4,4,4) = 0 which is indeed true.

I'm a little slow tonight. :smile:
 

FAQ: Express one vector as a linear combination of others

How can I express one vector as a linear combination of others?

The process of expressing one vector as a linear combination of others involves finding the coefficients or weights that when multiplied by each vector and added together, result in the desired vector. This can be done using systems of linear equations or matrix operations.

Why is it important to express vectors as linear combinations of others?

Expressing vectors as linear combinations of others allows us to represent a vector in terms of a set of basis vectors. This can simplify calculations and allow us to understand the relationship between different vectors.

How do I know if a vector can be expressed as a linear combination of others?

A vector can be expressed as a linear combination of others if it lies in the span of those vectors. This means that the vector can be reached by scaling and adding the given vectors together. If the vector lies outside the span, it cannot be expressed as a linear combination.

Can a vector be expressed as a linear combination of only two other vectors?

Yes, a vector can be expressed as a linear combination of only two other vectors as long as those two vectors are not linearly dependent. This means that one vector cannot be a multiple of the other. If the two vectors are linearly dependent, then the vector can be expressed as a linear combination of just one of them.

Can a vector be expressed as a linear combination of more than two other vectors?

Yes, a vector can be expressed as a linear combination of more than two other vectors. In fact, any vector in a vector space can be expressed as a linear combination of a set of basis vectors. The number of vectors needed in the linear combination will depend on the dimension of the vector space.

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