Linear independence

Definition: Linear independence. A subset is called linearly independent if for any implies that must be equal to 0 and zero only.

• Essentially, if there is no way to add the vectors in the set to get back to zero without multiplying by them zero, they are linearly independent.
• Examples of linearly independent vectors include any basis vectors, among others
  • Non-examples are sets of vectors where multiple are co-linear

Example: Say we have some . Suppose that . The only solution for this is . Thus, is linearly independent.

• There is a special set of linearly independent vectors:
  • They are vectors where the th element is 1 and the rest are zero.

Remark: If is linearly independent,

• This makes sense because any real number times zero equals zero
  • For a set of vectors to be linearly independent, they must reach zero through multiplying only by zero.

Proof: Suppose some and . Then is not linearly independent.

• This claims that if there is in , and we can make that vector through linear combinations of other vectors in the set, it is not linearly independent

As , then there exists . We want to see if can be made by these combinations. Since the coefficient of , and the total sums to 0, we can conclude that is not linearly independent.

Remark: If and is not linearly independent, then is also not linearly independent.

• This is just saying that is a subset of a set is not linearly independent, any larger sets that contain it are also not linearly independent.

Midterm review

• Say that is linearly independent. This means that
• Say that spans , then .