Did my Prof mark my answer wrong in error? (set of vectors: lin. independent)

[Karnbread]

Need verification on linear algebra question.

Question is "Explain what it means for a set of vectors S={v1,v2,v3....vk} to be linearly independent."

My answer was, "The set of vectors must have trivial solutions (no non trivial solutions). This means the linear combination of the span a1v1+a2v2....+akvk=0 will give us all a1,a2...ak = 0 such that ov1+0v2+...0vk=0. Thus linearly independent."

I found that the answer I inputted about the linear combinations was correct, but my professor marked it wrong. Can anyone tell me where I went wrong? Or is my answer right and he accidentally marked it wrong?

 

[pka]

Question is "Explain what it means for a set of vectors \(\displaystyle S=\{v_1,v_2,v_3....v_k\}\) to be linearly independent."
My answer was, "The set of vectors must have trivial solutions (no non trivial solutions). This means the linear combination of the span a1v1+a2v2....+akvk=0 will give us all a1,a2...ak = 0 such that ov1+0v2+...0vk=0. Thus linearly independent."

Well I don't think I would have marked it completely wrong, but is is extremely confusing to read.
Say if simply: The set \(\displaystyle S=\{v_1,v_2,v_3....v_k\}\) is linearly independent if an only if \(\displaystyle \alpha_1 v_1+\alpha_2 v_2 +\alpha_3 v_3....\alpha_k v_k=0\) does not have a non-trivial solution; i.e. \(\displaystyle (\forall j,~1\le j\le k)[\alpha_j= 0]\).

 

[Steven G]

Need verification on linear algebra question.

Question is "Explain what it means for a set of vectors S={v1,v2,v3....vk} to be linearly independent."

My answer was, "The set of vectors must have trivial solutions (no non trivial solutions). This means the linear combination of the span a1v1+a2v2....+akvk=0 will give us all a1,a2...ak = 0 such that ov1+0v2+...0vk=0. Thus linearly independent."

I found that the answer I inputted about the linear combinations was correct, but my professor marked it wrong. Can anyone tell me where I went wrong? Or is my answer right and he accidentally marked it wrong?

My answer was, The set of vectors must have trivial solutions (no non trivial solutions)
You should have said the set of vectors must ONLY have trivial solutions. In fact just say that a1v1+a2v2....+akvk=0 only has the trivial solutions, ie ai=o for i=1 to n

 

[HallsofIvy]

As I was reading your answer my first thought was "What does it even mean for a set of vectors to "have solutions"?

 

[Karnbread]

My answer was, The set of vectors must have trivial solutions (no non trivial solutions)
You should have said the set of vectors must ONLY have trivial solutions. In fact just say that a1v1+a2v2....+akvk=0 only has the trivial solutions, ie ai=o for i=1 to n

I get what you're saying but isn't stating no non trivial solutions the same as saying only trivial solutions? They both mean a1..a1..ak must be 0. Im a college student, not a math professor, my words may have been confusing but I feel as if my answer could have been concurred. Necessarily speaking, it was confusingly worded, but not wrong?

 

[Karnbread]

As I was reading your answer my first thought was "What does it even mean for a set of vectors to "have solutions"?

I meant that the linear combinations of the set of vectors must all be trivial solutions, meaning it must equal 0! I am not a math major my wording was strange but I wasn't wrong?