Linear algebra proof on linearly independence

In summary: For the second case, assume the vectors are NOT in the span of the preceding ones. Then prove they ARE linearly independent.In summary, the conversation discusses proving the linear independence of a set of vectors in a vector space V. It is shown that if the vectors have the properties that the first one is not equal to 0, and each subsequent vector is not in the span of the preceding ones, then the set is linearly independent. Conversely, if the set is an ordered list of linearly independent vectors, then it also has the above properties. The suggested approaches for proving this include using induction or proof by contradiction.
  • #1
gavin1989
8
0

Homework Statement


Show that if vectors v1 , . . . , vk in a vector space V have the properties that v1
does not = 0, and each vi is not in the span of the preceding ones, then the vectors are linearly independent.
Conversely, show that if v1 , . . . , vk is an ordered list of linearly independent vectors, then it has the above properties.

Homework Equations


The Attempt at a Solution



I know it is kinda easy to prove the set is linearly independent, but with the property there, how would I start the proof? I think since v2 is not a multiple of v1, and v3 is not a multiple of vi and v2 ... as well as v1 does not = 0. so i need to show: av1+bv2+cv3...nvn=0 becuz v1 does not = 0, and vi is not the span of the preceding ones(so they are all not =0), does it mean a=b=c...=0?

thanks
 
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  • #2
You would write a formal proof using induction. There's not much to prove for the k=1 case. Now assume it's true for k and show it's true for k+1. How would that proof look?
 
  • #3
is proof by induction the only way to do it? my prof has not really taught how to prove by induction
 
  • #4
Well, you could also do proof by contradiction. For the first case, assume the vectors are NOT linearly independent. Then prove there IS some vi that is in the span of the preceding ones.
 

1. What is linear independence in linear algebra?

Linear independence in linear algebra refers to a set of vectors in a vector space that cannot be expressed as a linear combination of other vectors in that space. In other words, no vector in the set can be written as a combination of the other vectors using scalar multiplication and addition. This concept is important in linear algebra because it allows us to determine whether a set of vectors can form a basis for a vector space.

2. How do you prove linear independence in linear algebra?

To prove linear independence in linear algebra, we need to show that the only solution to the equation c1v1 + c2v2 + ... + cnvn = 0, where c1, c2, ..., cn are scalars and v1, v2, ..., vn are vectors, is c1 = c2 = ... = cn = 0. This can be done by setting up a system of equations and using techniques such as Gaussian elimination or matrix operations to solve for the coefficients c1, c2, ..., cn. If the only solution is c1 = c2 = ... = cn = 0, then the set of vectors is linearly independent.

3. What is the importance of linear independence in linear algebra?

Linear independence is important in linear algebra because it allows us to determine whether a set of vectors can form a basis for a vector space. A basis is a set of linearly independent vectors that span the entire vector space. This means that any vector in the space can be written as a linear combination of the basis vectors. Bases are useful for many applications in mathematics and physics, such as solving systems of linear equations and representing transformations.

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

Yes, a set of two vectors can be linearly independent. In fact, any set of two non-zero vectors in a two-dimensional vector space is linearly independent. This is because two vectors in a two-dimensional space can never be scalar multiples of each other, which is a necessary condition for linear dependence. However, in higher-dimensional spaces, a set of two vectors may or may not be linearly independent, and this can be determined by solving the system of equations c1v1 + c2v2 = 0 and checking for non-trivial solutions.

5. How is linear independence related to linear dependence?

Linear independence and linear dependence are opposite concepts. If a set of vectors is linearly independent, then it is not linearly dependent. On the other hand, if a set of vectors is linearly dependent, then it is not linearly independent. In other words, a set of vectors is either linearly independent or linearly dependent, there is no in-between. Linear dependence occurs when one vector in a set can be expressed as a linear combination of the other vectors, which means it is not necessary for the set to span the entire vector space.

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