Linear Independence: Proving Dependence & Independence of Vectors, Sets

In summary, the conversation discusses the relationship between linear dependence and independence in a vector space. The first question states that two nonzero vectors, u and v, are linearly dependent if and only if one vector is a scalar multiple of the other. Similarly, they are linearly independent if neither vector is a multiple of the other. The second question states that a set of vectors, S, is linearly dependent if and only if one vector in S can be expressed as a linear combination of the other vectors in S. This must be proven in both directions. To approach these problems, the theorem or definition of linear independence should be used to prove the statements. Finally, the third question states that if S is a set of vectors in a vector
  • #1
hkus10
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1) Let u and v be nonzero vectors in a vector space V. show that u and v are linearly dependent if and only if there is a scalar k such that v = ku. Equivalently, u and v are linearly independent if and only if neither vector is a multiple of the other.

2) Let S = {v1, v2, ..., vk} be a set of vectors in a vector space V. Prove that S is linearly dependent if and only if one of the vectors in S is a linear combination of all the other vectors in S.

For these two questions, I know I have to prove them in both directions because of "if and only of". However, how to approach this problem? what Thms or definition should I use to prove them?

3) Let S = {v1, v2, ..., vk} be a set of vectors in a vector space V, and let W be a subspace of V containing S. Show that W contains span S.

For question 3, does "W be a subspace of V containing S" mean W contains S? If yes, what is the reason to show it?
 
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  • #2
What did you try already? What is your definition for linear independence?
 

1. What is linear independence?

Linear independence is a concept in linear algebra that refers to a set of vectors in a vector space. It means that none of the vectors in the set can be written as a linear combination of the others. In other words, no vector in the set can be expressed as a combination of the other vectors using scalar multiplication and addition.

2. How do you prove that a set of vectors is linearly independent?

To prove that a set of vectors is linearly independent, you can use the definition of linear independence. This means that you need to show that no vector in the set can be expressed as a linear combination of the others. You can do this by setting up a system of equations and solving for the coefficients of the linear combination. If the only solution is all coefficients equaling zero, then the set of vectors is linearly independent.

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

Yes, a set of two vectors can be linearly independent. As long as neither vector can be written as a linear combination of the other, the set is considered linearly independent. This is true for any number of vectors as long as none of them can be written as a linear combination of the others.

4. What is the difference between linear independence and linear dependence?

The difference between linear independence and linear dependence is that a set of vectors is linearly independent if none of the vectors can be written as a linear combination of the others. On the other hand, a set of vectors is linearly dependent if at least one vector can be written as a linear combination of the others. In other words, linear independence means that the vectors are "free" and not related to each other, while linear dependence means that there is some sort of relationship between the vectors.

5. Can a set of linearly dependent vectors still span a vector space?

Yes, a set of linearly dependent vectors can still span a vector space. This is because even though the vectors may be related to each other, they can still be used to form a basis for the vector space. However, the dimension of the vector space will be equal to the number of linearly independent vectors in the set, rather than the total number of vectors in the set.

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