Given

Given

1. A series of linear equations

   • m equations and n unknowns
   • May be written as Ax = b
     • A is an m x n matrix
     • x is an n x 1 vector
     • b is an m x 1 vector
     • Ax = b = 0
       • Homogenous system
         • Consistent
           • The linear combination of solutions is also a solution
         • x = 0 is the trivial solution
           • If there is 1 non-trivial solution, there are infinitely many
             • The null space is the set of all solutions
         • Given x as a solution
           • x + z is a solution for the inhomogenous system
     • Ax = b ≠ 0
       • Inhomogeneous system
         • Inconsistent
           • No solutions
         • Consistent
           • 1 solution
             • If Ax = 0 has 1 solution
           • Infinitely many solutions
             • If Ax = 0 has infinite solutions
       • Given y & z as solutions
         • y - z is a solution for the homogenous system
   • No solution
     • Inconsistent
   • Any solutions
     • Consistent
       • b is a linear combination of columns of A

2. A set of vectors

   • Scale & add vectors
     • Answer lies in the set
       • Closed
         • This is a vector space
     • This is a linear combination
       • The set of all possible linear combinations has many interpretations
         • All of the vectors in a plane
         • All of the points in a line
         • All of the vectors in a space
         • A point
     • Answer does not lie in the set
       • Not closed
         • Just a set
   • A subset of a vector space