Question

(15points) Let 21, 22, ..., Ik be linearly independ vectors in a vector space V. If we add a vector Ik+1 to the collection, will we still have a linear independent collection of vectors? Explain. If we delete a vector, say, 2k , from the collection, will we still have a linearly independent collection of vectors? Explain.

Answer 1

If we add a vector Ik+1 to the collection of 21, 22, ..., Ik, it is possible that the collection will no longer be linearly independent. Linear independence is determined by whether the vectors in the collection are capable of representing all vectors in the space. If Ik+1 can be expressed as a linear combination of 21, 22, ..., Ik, then the collection will not be linearly independent because Ik+1 can be represented by the other vectors in the collection.