Introduction

The span of a set of independent basis vectors is the set of all vectors that can be described as linear combinations of the basis vectors:

• One basis vector spans a line:

  

  is a 1D straight line through the origin.
• Two basis vectors span a plane:

  

  is a 2D plane through the origin.
• Three basis vectors span a 3D space:

  

  is a 3D space through the origin (in n-dimensional space.)

Remember the definition of independence: the basis vectors are independent if the only way to get zero is to take every c_i zero. It implies that you cannot express one basis vector in terms of the rest.

• Two vectors are linearly independent if they are not along the same line.
• Three vectors are linearly independent if they are not in the same plane.

The rank of a matrix is the number of independent rows, or columns. These are the same; see the example.