1)Suppose that the set S={v1,v2,…,vn} is a basis for the vector space V. Decide whether each of the following statements is true or false. If you do not have enough information to decide whether a statement is true or false,
1. The vectors v1,…,vn span V.
2. The vectors v1,…,vn are linearly independent.
3. If T={w1,w2,…,wm} is a different basis for V, then m could be equal to n+1.
4. If x is any vector in V, then it may be impossible to write x as a linear combination of elements of S.
6. The dimension of V is equal to n.

Question

1)Suppose that the set S={v1,v2,…,vn} is a basis for the vector space V. Decide whether each of the following statements is true or false. If you do not have enough information to decide whether a statement is true or false,
1. The vectors v1,…,vn span V.  
2. The vectors v1,…,vn are linearly independent.  
3. If T={w1,w2,…,wm} is a different basis for V, then m could be equal to n+1.  
4. If x is any vector in V, then it may be impossible to write x as a linear combination of elements of S.  
6. The dimension of V is equal to n.

inkyvoyd · Answer

you are missing 5 lol

inkyvoyd · Answer

1 and 2 are trivially true by definition of a vector space, basis, and spanning

isbella · Answer

thanks this is 5. There exist nonzero constants c1,…,cn such that c1v1+⋯+cnvn=0

inkyvoyd · Answer

I'm pretty sure 5 is false by definition of linearly independent, but I'm not too sure I remember

inkyvoyd · Answer

http://en.wikipedia.org/wiki/Linear_independence#Definition

isbella · Answer

3 4 are false right and 6 true ?

inkyvoyd · Answer

4 is false, I don't know the answer to 3 lol

inkyvoyd · Answer

OH, yeah, 3 is false

inkyvoyd · Answer

and yeah, looks like 6 is true.

inkyvoyd · Answer

you might want to look over these answers with another person if possible though