Linear Combinations, Span, and Basis Vectors

Vector Coordinates as Scalars

  • When describing a vector with coordinates (e.g., (3, -2)), think of each coordinate as a scalar that stretches or squishes vectors.
  • In the x-y coordinate system, two special vectors are:
    • i-hat: Unit vector in the x-direction, pointing to the right with length 1.
    • j-hat: Unit vector in the y-direction, pointing straight up with length 1.
  • The x-coordinate scales i-hat, and the y-coordinate scales j-hat. The vector described by the coordinates is the sum of these scaled vectors.

Basis of a Coordinate System

  • i-hat and j-hat together form the basis of a coordinate system.
  • Basis vectors are what the scalars scale when thinking about coordinates as scalars.
  • We could choose different basis vectors, resulting in a different coordinate system.
  • Anytime vectors are described numerically, it depends on the implicit choice of basis vectors.

Linear Combination

  • Scaling two vectors and adding them together is called a linear combination of those two vectors.
  • If one scalar is fixed and the other changes freely, the tip of the resulting vector draws a straight line.
  • If both scalars range freely:
    • For most pairs of vectors, every point in the plane can be reached.
    • If the two original vectors line up, the tip of the resulting vector is limited to a single line through the origin.
    • If both original vectors are zero vectors, you are stuck at the origin.

Span

  • The span of two vectors is the set of all possible vectors reachable via a linear combination of those vectors.
  • The span of most pairs of 2D vectors is all of 2D space.
  • If two vectors line up, their span is all vectors whose tips lie on a certain line.
  • The span of vectors indicates all possible vectors reachable through vector addition and scalar multiplication.

Vectors as Points

  • When dealing with collections of vectors, it is common to represent each vector by a point in space (the point at the tip of the vector with its tail at the origin).
  • Thinking about every possible vector whose tip is on a certain line is equivalent to thinking about the line itself.
  • Considering all possible 2D vectors is equivalent to thinking about the infinite flat sheet of 2D space.
  • Think of a vector as an arrow when considering it on its own, and as a point when dealing with a collection of vectors.
  • The span of most pairs of vectors is the entire infinite sheet of 2D space.
  • If the vectors line up, their span is just a line.

Span in Three-Dimensional Space

  • The span of two vectors in 3D space (not pointing in the same direction) is the collection of all linear combinations of those vectors.
  • The tip traces out a flat sheet cutting through the origin of 3D space.
  • This flat sheet is the span of the two vectors.
  • A linear combination of three vectors involves choosing three scalars, scaling each vector, and adding them.
  • The span of three vectors is the set of all possible linear combinations:
    • If the third vector lies on the span of the first two, the span doesn't change, and you're trapped on the same flat sheet.
    • If the third vector doesn't lie on the span of the first two, it unlocks access to every possible 3D vector.

Redundancy and Linear Dependence

  • If one of the vectors is redundant and doesn't add anything to the span.
  • Whenever one of the vectors can be removed without reducing the span, there is a terminology to describe this.