Proof about linear independence

In summary, linear independence refers to a set of vectors that cannot be expressed as a linear combination of each other. To prove linear independence, one must show that the only solution to the equation c1v1 + c2v2 + ... + cnvn = 0 is when all the constants are equal to 0. This is important in fields such as mathematics, physics, and engineering as it allows us to determine if a set of vectors can serve as a basis for a vector space. A set of linearly dependent vectors cannot be linearly independent, and there is a difference between linear independence and orthogonality, with the former being a necessary but not sufficient condition for the latter.
  • #1
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suppose v1,...,vk are nonzero vectors with the property that vi.vj=0 whenever i is not equal to j. Prove that {v1,...,vk} is linearly independent.
 
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  • #2
Suppose [tex]\alpha_1, \ldots, \alpha_k[/tex] are such that [tex]0 = \alpha_1 v_1 + \ldots + \alpha_k v_k[/tex]. Try taking the dot product of this equation with each of the [tex]v_i[/tex]s and see what it tells you about the [tex]\alpha_i[/tex]s.
 
  • #3
thank u, i got it now...
 

Related to Proof about linear independence

1. What is linear independence?

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

2. How do you prove linear independence?

To prove linear independence, you must show that the only solution to the equation c1v1 + c2v2 + ... + cnvn = 0, where c1, c2, ..., cn are constants and v1, v2, ..., vn are the vectors in the set, is when all the constants are equal to 0. This can be done using various methods such as Gaussian elimination, determinants, or the definition of linear independence.

3. What is the importance of proving linear independence?

Proving linear independence is important in various fields such as mathematics, physics, and engineering. It allows us to determine if a set of vectors can serve as a basis for a vector space, which is crucial in solving systems of linear equations and understanding the properties of vector spaces.

4. Can a set of linearly dependent vectors be linearly independent?

No, a set of linearly dependent vectors cannot be linearly independent. If a set of vectors is linearly dependent, it means that at least one vector in the set can be expressed as a linear combination of the other vectors. This violates the definition of linear independence, which requires each vector to be independent of the others.

5. What is the difference between linear independence and orthogonality?

Linear independence refers to a set of vectors that cannot be written as a linear combination of each other, while orthogonality refers to a set of vectors that are perpendicular to each other. Linear independence is a necessary but not sufficient condition for orthogonality. In other words, a set of linearly independent vectors can be orthogonal, but a set of orthogonal vectors may not be linearly independent.

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