Show that in every set p2 with more than three vectors is linearly dependent.

In summary, linear dependence in linear algebra refers to a set of vectors where one or more can be expressed as a linear combination of the others. It is important to prove linear dependence in sets with more than three vectors as it provides valuable insights into the relationships and dependencies among the vectors. The significance of having more than three vectors in a set is that it increases the likelihood of linear dependence. Methods such as Gaussian elimination and solving for coefficients can be used to show linear dependence in a set with more than three vectors. It is not possible for a set with more than three vectors to be linearly independent as it can have at most n-1 linearly independent vectors, meaning at least one vector must be linearly dependent.
  • #1
delgeezee
12
0
i know S = { \(\displaystyle 1 , x, x^2\)} is linearly dependent set for p2. where \(\displaystyle (a_0, a_1, a_2) = (0,0,0) \)
I wanted to use the Wronskian on { \(\displaystyle 1 , x, x^2, x^3\)} , but as I understand, it only proves linear independence and not the converse.
 
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  • #2
delgeezee said:
i know S = { \(\displaystyle 1 , x, x^2\)} is linearly dependent set for p2. where \(\displaystyle (a_0, a_1, a_2) = (0,0,0) \)
I wanted to use the Wronskian on { \(\displaystyle 1 , x, x^2, x^3\)} , but as I understand, it only proves linear independence and not the converse.

Hi delgeezee!

Can you elaborate?
For starters, what do you mean by p2?
And how do your $a_i$ tie in?
 

Related to Show that in every set p2 with more than three vectors is linearly dependent.

What is the meaning of "linearly dependent" in this context?

In linear algebra, a set of vectors is considered linearly dependent if one or more of the vectors can be expressed as a linear combination of the others. This means that at least one of the vectors is not necessary to span the entire set.

Why is it important to show that a set with more than three vectors is linearly dependent?

Proving that a set is linearly dependent is important because it allows us to understand the relationships and dependencies among the vectors in the set. This information can be useful in solving problems and making predictions in various branches of science and mathematics.

What is the significance of having more than three vectors in the set?

Having more than three vectors in a set increases the likelihood of linear dependence. This is because as the number of vectors increases, so does the number of possible combinations and relationships among them. Therefore, showing that a set with more than three vectors is linearly dependent can provide valuable insights into the structure and behavior of the vectors.

What methods can be used to show that a set with more than three vectors is linearly dependent?

One method is to use Gaussian elimination, which involves performing row operations on a matrix composed of the vectors. If the resulting matrix has a row of zeros, then the set is linearly dependent. Another method is to use the definition of linear dependence and solve for the coefficients of the linear combination for each vector. If at least one coefficient is not equal to zero, then the set is linearly dependent.

Can a set with more than three vectors ever be linearly independent?

No, a set with more than three vectors cannot be linearly independent. This is because a set with n vectors can have at most n-1 linearly independent vectors. Therefore, if a set has more than three vectors, there must be at least one linearly dependent vector.

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