Linear Independence
Suppose u and v are two vectors in such that is a plane through the origin. Now we add another vector w in the mix. There are two possible outcomes for : Either the same plane through the origin or a set properly containing the plane (Note: in this case, it is actually the whole ). How can we know which outcome we have? This leads us to an important notion in linear algebra called linear independence.
First case: Suppose . Then it means that you cannot get any new vectors by taking linear combinations of u, v, and w. In particular, w must be in I.e. there exist such that .
When this happens, we say that u, v, and w are linearly dependent . More generally, for any vectors , they are linearly dependent if there exists at least one vector such that it is a linear combination of the remaining vectors.
However, such definition is not very convenient to use because usually we do not know in advance exactly which vector is a linear combination of others. And by the above example, we can rewrite the condition as follows: . (Note: Here “0” means the zero vector.). Therefore, we use the following equivalent definition which is much more useful:
Let be set of vectors in . Then the set is linearly dependent if there exist such that at least one of them is non-zero and .
Question: Why the above definition is equivalent to saying that there exists a vector in that is a linear combination of the others?
Second case: Suppose . Then it means that new vectors are generated by taking linear combinations of u, v, and w. In other words, u, v, and w are NOT linearly dependent. Any set that is not linearly dependent is linearly independent. More generally, we have the following definition:
Let be set of vectors in . Then the set is linearly independent if for any such that , then .
Heuristically speaking, a set of vectors is linearly independent means that there is no redundant vector in the set i.e. if you throw away any one of them, the span will be smaller.
The following are some T/F questions that test your understanding of the concepts:
If u, v, and w are vectors in such that , then are linearly dependent.
For any vector v in , is linearly independent.
Any set of vectors in that contains zero vector is linearly dependent.
For any vectors u and v in , if is linearly dependent, then either u is a scaling of v or v is a scaling of u.
For any vectors u and v in , if is linearly dependent, then is a line through the origin.