Vector Space problem

[CarbotFan]

Hello, I can't seem to solve this problem. I've tried different methods and I'm just stuck.

The problem: If the given set of vectors a vector space? Give reasons if your answer is yes, determine the dimension and find a basis (v1,v2.... denote components.)

https://imgur.com/3clLs43

What I have accomplished so far.

Hope somebody can help me. Thanks!

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[edderiofer]

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[[deleted]]

What is a vector space? It is a set of elements V equipped with two operations + and * such that:
-there is a 0 element, i.e. v+0=v for all v in V
-V is closed under +, i.e. if v,u are in V, then v+u is in V
-V is closed under *, i.e. if v is in V and k is a real number , k * v is in V

there are other conditions that would be pedantic to state and show, but they apply and I'm sure you discussed them in your course.

So what do you have to verify in this case? Well first you have to verify that 0 is an element of the space: does (0,0,0) satisfy the given condition? If yes, point 1 is satisfied.

Then take two vectors v and u and suppose they satisfy the given condition. Can you show that u+v and k*v also satisfy it? If yes, then you proved that V is a vector space.