Suppose B = {V1, ..., V6} is a set of six vectors in R4. (Note that no further information on the six vectors is provided.) Each of the statements listed below contains three options: select the correct option and give evidence of your thinking. For example, if a statement is always true, you should explain why. If a statement may be true, you should provide examples. 1. The vectors (do) (do not) (might not) span R4.2. The vectors (are) (are not) (may be) linearly independent. 3. Any four of these vectors (are) (are not) (may be) a basis for R4.

Step-by-step explanation:

1) The vectors might no span R^4. if you want to span R^4, at least 4 vector of the six must be linear indepent. But, consider the following set of vectors

{ (1,1,1,1), (2,2,2,2), (3,3,3,3), (4,4,4,4), (5,5,5,5), (6,6,6,6) }. This set has 6 vectors of R4, but they are all linearly dependent, since they are a linear combination of the vector (1,1,1,1). Since we have only one linear indepent vector, we cannot span R4 with this set.

2) Since R4 is a 4-dimensional vector space, by having 6 vectors in an 4-dimensional space, the must necessarily be linearly dependent, since in R4 the maximum number of linearly independent vectors we could have in a set is 4.

3). The statement may be true. Any four subset of these vectors may be a base. To do so, that depends on their linear independence. If they are linearly independence, they are base since they are 4 linearly independent vectors in a 4-dimensional space, but if they lack linearly independence, they cannot be a base.

For example, consider the following set of vectors

{ (1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1), (1,1,1,1), (0,0,0,1) }.

Note that if we take the first 4 vectors, they are linearly indepent, hence they are a base. Same happens if we take vectors 2-5. But if we take vectors 3-6, then this subset is not a base, since it has only 3 linearly independent vectors.