27. Span

Represents the subspace of the vector space that is “covered” by these vectors through their linear combinations

If you have a set of vectors , the span of these vectors is the set of all vectors that can be written as:

Any vector in can be represented by a linear combination with some combination of these vectors

Example

1. Spanning

Any point can be represented as a linear combination of and

  1. Define the vectors
  1. Express as a linear combinations

Which expands to

  1. Set up the system of equations
  1. Express : From equation (1), we can directly express
  1. Substitute into equation (2)

Rearranging gives:

  1. Solve for : Dividing both sides by 3 yields
  1. Example with specific values: Let’s say we want to find and when

Substitute and

  1. Final linear combination: Now, substituting and back into the linear combination

Verifying

  1. This shows that

2. Spanning Line in

Any linear combination of and will produce vectors that lie along the same line. This is the line through the origin in the direction of (or ), with all points on the line being scalar multiples of