Summary of Linear combinations and span | Vectors and spaces | Linear Algebra | Khan Academy

Main Ideas and Concepts

• Linear Combinations:

  A linear combination of Vectors involves adding Vectors together, each multiplied by a scalar (real number). For Vectors v₁, v₂, …, v_n, a linear combination can be expressed as:

  c1 v1 + c2 v2 + … + cn vn

  Here, c₁, c₂, …, c_n are arbitrary constants from the set of real numbers.

• Example of Linear Combinations:

  Given Vectors a = (1, 2) and b = (0, 3):

  • A linear combination could be 3a - 2b, resulting in the vector (3, 0).
• Visual Representation:

  The Vectors can be visualized in a coordinate system, demonstrating how different Linear Combinations can produce various resultant Vectors.

• Span of Vectors:

  The Span of a set of Vectors is the set of all possible Vectors that can be formed through Linear Combinations of those Vectors.

  • For Vectors a and b, if they are not collinear, their Span in &mathbb;R² is all of &mathbb;R².
  • If Vectors are collinear (e.g., a = (2, 2) and b = (-2, -2)), their Span is limited to a line, not the entire plane.
• Special Case of the Zero Vector:

  The Span of the Zero Vector is just the Zero Vector itself, as all Linear Combinations will yield the Zero Vector.

• Basis and Orthogonality:

  The unit Vectors i = (1, 0) and j = (0, 1) are orthogonal and form a Basis for &mathbb;R², meaning any vector in this space can be represented as a linear combination of i and j.

• Formal Definition of Span:

  The Span of Vectors v₁, v₂, …, v_n is defined as:

  Span(v1, v2, …, vn) = { c1 v1 + c2 v2 + … + cn vn | ci ∈ &mathbb;R }

Methodology and Instructions

• To find the linear combination of Vectors:
  • Identify the Vectors involved (e.g., a and b).
  • Choose arbitrary constants (scalars) to multiply each vector.
  • Add the scaled Vectors together to get the resultant vector.
• To determine the Span of Vectors:
  • Assess if the Vectors are collinear.
  • If they are not, conclude that their Span covers the entire space (e.g., &mathbb;R²).
  • If they are collinear, the Span is limited to a line.
• To express any point in &mathbb;R² as a linear combination:
  • Set up equations based on the linear combination of the Vectors.
  • Solve for the coefficients to express the desired point.

Speakers or Sources Featured

• Sal Khan (Khan Academy)

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