Independent or Dependent?
Independent and Dependent Vectors
But the manifestation of the Spirit is given to each one for the profit of all: for to one is given the word of wisdom through the Spirit, to another the word of knowledge through the same Spirit, to another faith by the same Spirit, to another gifts of healings by the same Spirit, to another the working of miracles, to another prophecy, to another discerning of spirits, to another different kinds of tongues, to another the interpretation of tongues. But one and the same Spirit works all these things, distributing to each one individually as He wills. For as the body is one and has many members, but all the members of that one body, being many, are one body, so also is Christ. For by one Spirit we were all baptized into one body— whether Jews or Greeks, whether slaves or free—and have all been made to drink into one Spirit. For in fact the body is not one member but many. If the foot should say, “Because I am not a hand, I am not of the body,” is it therefore not of the body? And if the ear should say, “Because I am not an eye, I am not of the body,” is it therefore not of the body? If the whole body were an eye, where would be the hearing? If the whole were hearing, where would be the smelling? But now God has set the members, each one of them, in the body just as He pleased. And if they were all one member, where would the body be? But now indeed there are many members, yet one body. And the eye cannot say to the hand, “I have no need of you”; nor again the head to the feet, “I have no need of you.” No, much rather, those members of the body which seem to be weaker are necessary. And those members of the body which we think to be less honorable, on these we bestow greater honor; and our unpresentable parts have greater modesty, but our presentable parts have no need. But God composed the body, having given greater honor to that part which lacks it, that there should be no schism in the body, but that the members should have the same care for one another. And if one member suffers, all the members suffer with it; or if one member is honored, all the members rejoice with it.
One way to demonstrate that a collection of vectors is dependent is to show that one vector in the set can be written as a nontrivial linear combination of other vectors in the set. In this case, the one vector can be eliminated from the set without loss of any vectors from the span of the set. If eliminating any single vector from the set changes the span of the set, then that vector is not a linear combination of the remaining. Students quickly learn that a set of vectors cannot be dependent and independent at the same time--if a set of vectors is independent, there is no reason to continue searching for a way to eliminate a vector from the set. All are essential in that case.
Christians form an independent collection in the church, too. Each person is given a unique set of spiritual gifts that is essential in completing the church community. However, when you read 1 Corinthians 12, it is quickly apparent that this essential to the whole type of independence does not mean "independent" in a colloquial sense of the word--each is still dependent upon all the others in the church community to fulfill the mission God has ordained for that group. To create the same "span," each individual is needed and all need the others. Again, all are essential!
