Linear Independence

[hkus10]

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?

Edited March 21, 2011 by hkus10

[DrRocket]

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?

 

 

For 1 & 2 all that you need is the definition of linear independence.

 

For 3 you need the definitions of "span" and of "vector space".

 

The statement "let W be a subspace of V containing S' means W contains S and W is a subspace of V.