linear independence

linear dependent

• the state that a vector can be written by other vectors
• eg. we have $a,b,c$ are vectors; $c=a+2b$; ⇒ $2a+b-c=2a+b-(a+2b)$ ⇒ a, b, c are linearly dependent

linear independent

• none of the vectors can be written in terms of other vectors
• require that the linear combination equation is equal to $\vec{0}$ when all scalars are equal zero
  • A: $c_1\overrightarrow{v_1}+c_2\overrightarrow{v_2}+\cdots +c_n\overrightarrow{v_n}=\vec{0}$
• check independence (normal, slow way)
  • find a valid set of nonzero scalars ($c_{1}to{c}_n)$ satisfying A
    • if possible → linear dep
    • if impossible → linear indep

prove linear independence

• Way 1: (elementary row operations) find scalars to see whether all of them = 0 when scalars = 0
  • just like in check independence above
• Way 2: row echelon
  • do elementary row operations so that diagonal matrix have diagonal value of 1, the below diag of 0.
  • if after construction → REF is 100% match or u cannot write any vector in terms of the other → linear indep
  • 
  • 

• Way 3: finding determinant (sqr matrix only)
  • if zero → linear dep
  • if nonzero → linear indep