Question

let be a set of vectors in with . select the best statement. note you only have 3 attempts. a. the set is linearly independent, as long as it does not include the zero vector. b. the set is linearly dependent. c. the set could be either linearly dependent or linearly independent, depending on the case. d. the set is linearly independent. e. the set is linearly independent, as long as no vector in is a scalar multiple of another vector in the set. f. none of the above

Answer 1

The correct answer to whether a set of vectors is linearly dependent or independent is that it is linearly independent if no vector is a scalar multiple of another and the zero vector is not included.

Step-by-step explanation:

The question concerns whether a set of vectors in linear algebra is linearly dependent or independent. For a set to be linearly independent, none of the vectors can be written as a linear combination of the others. This means that:

• The set is linearly independent if and only if the only solution to the equation a1v1 + a2v2 + ... + anvn = 0 is a1 = a2 = ... = an = 0, where vi are vectors in the set and ai are scalars.

Statement (e) 'the set is linearly independent, as long as no vector in is a scalar multiple of another vector in the set' aligns with the definition of linear independence, as long as we also ensure that the zero vector is not included in the set. Hence, the correct answer would be:

• (e) The set is linearly independent, as long as no vector in the set is a scalar multiple of another vector in the set and it does not include the zero vector.