Linear Independence
For some sets of vectors v1,...,vn, a single vector can be written in two different ways as a linear combination of them:
Equivalently, by subtracting these a non-trivial combination is zero:
If that is possible, then v1,...,vn are called linearly dependent; otherwise, they are linearly independent. Similarly, we can speak of linear dependence or independence of an arbitrary set S of vectors.
If S is linearly independent and the span of S equals V, then S is a basis for V.
Read more about this topic: Linear Combination
Famous quotes containing the word independence:
“We commonly say that the rich man can speak the truth, can afford honesty, can afford independence of opinion and action;—and that is the theory of nobility. But it is the rich man in a true sense, that is to say, not the man of large income and large expenditure, but solely the man whose outlay is less than his income and is steadily kept so.”
—Ralph Waldo Emerson (1803–1882)