Question

Sis a neatly dependent set, then each vector is a linear combination of the other vectors in S. Choose the correct answer below

A. True: If an indexed set of vectors, is nearly dependent, then at least one of the vectors can be written as a linear combination of other vectors in the set. Using the basic properties of equality, each of the vectors in the near combination can also be writen as a linear combination of those vectors
B True. If is nearly dependent then for each va vector in S, is a linear combination of the preceding vectors in S
C. False HS is nearly dependent then there is a stone vector that it not a near combination of the other vectors, but the others may be linear combination of each other
D . Fabe an indexed set of vectors, S. is linearly dependent, then it is only necessary that one of the vectors is a linear combination of the other vectors in the st

Answer 1

The correct answer is A. True: If a set S of vectors is linearly dependent, at least one vector is a linear combination of the others, which confirms that not every vector needs to be a combination of all the others.

Step-by-step explanation:

If a set S is linearly dependent, then at least one of the vectors in the set can be written as a linear combination of the other vectors in the set. This means there are scalars, not all of which are zero, such that their linear combination equals zero. Therefore, if we rearrange these terms, we can express one vector as a linear combination of the others. However, it is not necessary that each vector in S is a linear combination of the other vectors. It only requires one vector to satisfy this condition to make the entire set linearly dependent.

The correct answer to the question is option A: True: If an indexed set of vectors S is linearly dependent, then at least one of the vectors can be written as a linear combination of other vectors in the set. It is not necessary that each vector is a linear combination of all the preceding vectors or any specific subset of vectors in S.