Question

let be a set of linearly dependent vectors in . select the best statement. note you only have 3 attempts. a. the set must span . b. the set spans , as long as it does not include the zero vector. c. the set spans , as long as no vector in is a scalar multiple of another vector in the set. d. the set does not span if some vector in is a scalar multiple of another vector in the set. e. the set cannot span . f. the set could, but does not have to, span . g. none of the above

Answer 1

The best statement is that a set of linearly dependent vectors could, but does not have to, span R^n. Linear dependence doesn't guarantee failure to span, as the set could still have enough vectors to cover the entire space despite redundancy.

Step-by-step explanation:

The question asks about linearly dependent vectors in a vector space and which statement best describes their properties regarding their ability to span the space. Linearly dependent means that at least one vector in the set can be written as a combination of others.

This can mean that the set may not span the entire space, especially if the vectors are scalar multiples of each other. But it's not just about including the zero vector or scalar multiples; the critical point is that for a set to span a space, it must contain enough vectors to cover the entire space, and these vectors must be linearly independent.

Therefore, the best statement is (f) the set could but does not have to, span R^n. This reflects that while linear dependence implies there's some redundancy and possibly a failure to span the entire space, it's not automatically guaranteed that the set cannot span the space—it could contain enough vectors to do so despite some being redundant.