Linear Independence

Linear Independence

1. Linear Dependent

   • Contains the zero vector
   • some vi can be expressed as a linear combination of the others
   • If a collection of vectors is larger than dim V ( the size of the basis) i

2. only if no vi can be expressed a linear combination of the others

3. nonzero rows in a reduce matrix form

   • Form a basis for the row space of E and every matrix row-equivalent to E

4. if the rREF has no zero rows

5. det not equal to zero

6. BASIS

   • Spanning Set
     • A subset of that is a basis
   • every vector in B can be expressed in one way as a linear combination of the vectors in B
   • Reduced echelon form is I
   • Det not equal to zero
   • Vectors spaces always have
   • Proof of vector
     • spanning set
     • LI
     • contains dim V vectors
   • Every symmetric matrix has a basis consisting of orthogonal eigenvectors

7. You can add vectors to produce a basis