Linear Algebra true or false

[kyle123]

please check my work im not sure for the last ones,,
True or False, only True if it is always true
1) Any linearly independent set of three vectors in R^3 is a basis for R^3___true

2) Any set of five vectors in R^4 spans R^4__false

3) Any set of four vectors in R^3 is linearly independent__false

4) the set {(1,0,1),(-5,4,-9),(5,-3,8),(2,-1,3)} spans R^3___

5) the set {x^3 – x + 2, x^2 + x – 2, 3x^3 + 2x^2 + 4x} spans P^3___

6) If a vector space V has a basis S with 7 elements, then any other basis T for V also has

7 elements____true

7) If a set S of vectors in V contains the zero vector, then S is linearly dependent___true

8) If dim(V) = n, then any set of n – 1 vectors in V must be linearly independent___

9) if dim(V) = n, then any set of n + 1 vectors in V must be linearly independent____

10) if dim(V) = n, then there exists a set of n + 1 vectors in V that spans V____



can u please help me on this problem,,,
5) Let V be any vector space
(a) prove that for all ν ε V and all scalars c, if cν=0 and c ≠ 0, then υ = 0
each step must be justified either by an axion or a theorem.
thanks

 

[daon]

For 8, consider n-1 vectors that are multiples of eachother. Are they linearly independent?

Similarly for 9.

For 10, consider a basis B for V. It has dimension n, and it spans V. Will throwing in another vector in V with the set B affect the sets span?

 

[kyle123]

ohhh

so is 8 true and 9 false, and 10 true,,, are the other ones right??

oh do u have any ideas of how to do five 5) Let V be any vector space
(a) prove that for all ν ε V and all scalars c, if cν=0 and c ≠ 0, then υ = 0
each step must be justified either by an axion or a theorem.
thanks for ur help

 

[daon]

I urge you to look again at what I said for 8. Granted this must give you the answer, but I'm wondering how you came to your conclusion. For example, for any vector v in V, {v,2v,3v,...,(n-1)v} is almost as dependent a set of n-1 vectors can possibly be.

The others you answered looked fine.

As for your proof, I do not have your list of axioms. It seems as simple as multiplying through by \(\displaystyle \frac{1}{c}\) (assuming it is defined over some field like the reals).