linearly dependent

(Minerva) (CliffNotes) (3Blue1Brown)

A set of vectors is linearly dependent if at least one of the vectors is a linear combination of the others. The set is linearly independent otherwise.

Example

a linear combination of the other two. So set is linearly dependent

Note:

1. Any collection of vectors  that contains the zero vector is automatically linearly dependent
2. The concept of linear (in)dependence works on both finite and infinite number of vectors

To find out whether a set of vectors is linearly dependent, prove that it is NOT linearly independent

Create a linear combination out of the vectors and equate that to the zero vector

Convert it to a linear system

Solve for linear system using Gaussian elimination to obtain reduced row echelon form Obtain solutions for . If , the set is linearly independent; otherwise linearly dependent