Independence:
The columns of A are independent when the nullspace N (A) contains only the zero vector.
Example1:
1. If three vectors are not in the same plane, they are independent. No combination of V1, V2, V3 in Figure 3.4 gives zero except 0V1 + 0V2 + 0V3.
2. If three vectors W1, W2, W3 are in the same plane, they are dependent.
Example2:
Vectors that Span a Subspace:
A basis of a vector space:
Every vector v in the space is a combination of the basis vectors, because they span the space.There is one and only one way to write v as a combination of the basis vectors.
Dimension:
The dimension of C(A) is the rank of matrix A. The dimension of N(A) is the number of free variables(n-r)!