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 𝑣1,𝑣2,…,𝑣𝑛, the span of these vectors is the set of all vectors that can be written as:

Span(𝑣1,𝑣2,…,𝑣𝑛)={𝑐1𝑣1+𝑐2𝑣2+…+𝑐𝑛𝑣𝑛|𝑐1,𝑐2,…,𝑐𝑛∈ℝ}

Any vector in ℝ2 can be represented by a linear combination with some combination of these vectors

Example

1. Spanning ℝ2

Span(𝑎⃗,𝑏⃗)=ℝ2𝑎⃗=[12]𝑏⃗=[03]3𝑎⃗+(−2)𝑏⃗=[3−06−6]=[30]

Any point 𝑥⃗ can be represented as a linear combination of 𝑎⃗ and 𝑏⃗

  1. Define the vectors
𝑎⃗=[12]𝑏⃗=[03]𝑥⃗=[𝑥1𝑥2]
  1. Express 𝑥⃗ as a linear combinations
𝑐1𝑎⃗+𝑐2𝑏⃗=𝑥⃗

Which expands to

𝑐1[12]+𝑐2[03]=[𝑥1𝑥2]
  1. Set up the system of equations
1𝑐1+0𝑐2=𝑥1(1)2𝑐1+3𝑐2=𝑥2(2)
  1. Express 𝑐1: From equation (1), we can directly express 𝑐1
𝑐1=𝑥1
  1. Substitute 𝑐1 into equation (2)
2𝑥1+3𝑐2=𝑥2

Rearranging gives:

3𝑐2=𝑥2−2𝑥1
  1. Solve for 𝑐2 : Dividing both sides by 3 yields
𝑐2=𝑥2−2𝑥13
  1. Example with specific values: Let’s say we want to find 𝑐1 and 𝑐2 when 𝑥⃗=[22]

Substitute 𝑥1=2 and 𝑥2=2

𝑐1=𝑥1=2𝑐2=2−2⋅23=−23
  1. Final linear combination: Now, substituting 𝑐1 and 𝑐2 back into the linear combination
2𝑎⃗−23𝑏⃗=[22]

Verifying

2[12]+13[03]=[22]
  1. This shows that
2𝑎⃗−23𝑏⃗=𝑥⃗

2. Spanning Line in ℝ2

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

𝑎⃗=[22]𝑏⃗=[−2−2]3𝑎⃗+(−2)𝑏⃗=[6−46−4]=[22]