Important Definitions and Theorems From Chapter 4

1. Definition: A subset of a vector space V is a subspace if

   1. u and v in V implies that u + v is in V

   2. c a real number and u in V implies that cu is in V

2. Definition:  v₁, v₂, ..., v_k are linear independent if 
       c_k1v₁ + c₂v₂ + ...+  c_kv_k = 0 
   implies that each of the c_i are zero.

3. Definition:  v₁, v₂, ..., v_k span the vector space V  if any v in V can be written as a linear combination of the v_i.
       c_k1v₁ + c₂v₂ + ...+  c_kv_k = V

4. Theorem:  If  v₁, v₂, ..., v_k are linearly dependant then at least one of the v_i can be written as a linear combination of the rest of the v's.

5. Definition:   v₁, v₂, ..., v_k are a basis for V is they are linear independent and they span V.

6. Theorem:  Let S = {v₁,v₂ ..., v_k} span a vector space V, then some subset of S is a basis for V.

7. Definition:  The dimension of a vector space V is the number of elements in a basis for V.

8. Theorem:  If T = {v₁, v₂, ..., v_r} is a set of linearly independent vectors in vector space of dimension n, then r < n.

9. Theorem:  If S is a set of linearly independent vectors in finite dimensional vector space V then there is a basis T of V that contains V.

10. Theorem:  If Dim(V) = n and S  is a set of vectors in V, then the following are equivalent:

    1. S is a basis for V.

    2. S is linearly independent  

    3. S spans V

11. Definition:  The null space of a matrix A is the set of vectors x with Ax = 0.  The dimension of the null space is called the nullity of A

12. Definition:  The row space of a matrix A is the span of the rows of A.

13. Theorem:  If A and B are row equivalent then they have the same row space.

14. Definition:  The column space of a matrix A is the span of the columns of A.

15. Theorem/Definition:  For any matrix A the dimension of the row space equals the dimension of the column space.  This dimension is called the rank of A

16. Theorem:  If A is an m x n matrix, then rankA + nullityA = n

17. Theorem:  Let A be an n x n matrix.  Then the following are equivalent:

    1. A is nonsingular.

    2. det(A) is nonzero.

    3. Ax = b has a unique solution for every vector b.

    4. Ax = 0 has only the trivial solution.

    5. rank(A) = n.

    6. A is row equivalent to the identity.

    7. A had nullity 0.

    8. The rows of A are linearly independent.

    9. The columns of A are linearly independent.

18. Theorem:  Ax = 0 the corresponding matrix equation of m equations and n unknowns has a nontrivial solution if and only if rank(A) < n.

19. Theorem:  Ax = b has a unique solution if and only if rank(A) = rank(A|b).  (the augmented matrix)