Linear combinations, span, and basis vectors | Chapter 2, Essence of linear algebra

Vector Coordinates and Linear Algebra

• Vector coordinates are numeric pairs that describe vectors, such as (3, -2).

• Each coordinate can be viewed as a scalar that stretches or squishes vectors.

Special Vectors in 2D

• In the xy coordinate system:

  • i-hat: unit vector pointing right (length = 1 in x direction)

  • j-hat: unit vector pointing up (length = 1 in y direction)

• The x coordinate of a vector scales i-hat, and the y coordinate of a vector scales j-hat.

Vector Representation

• The vector represented by (3, -2) is:

  • 3 times i-hat (stretched to the right)

  • -2 times j-hat (flipped and stretched downwards)

• This describes the sum of two scaled vectors.

Basis of a Coordinate System

• Basis Vectors: i-hat and j-hat are the basis vectors of the standard coordinate system.

• Basis vectors allow for a linear combination of vectors by scaling.

• Choosing different basis vectors leads to different coordinate systems, enabling different 2D vectors through scalar selection.

Linear Combinations

• A linear combination involves scaling two vectors and adding them together.

• The term "linear" relates to the ability to draw a straight line when fixing one scalar and changing another;

  • Tips of resultant vectors trace lines through the origin.

Span of Vectors

• The span of vectors refers to all possible vectors created by linear combinations of the original vectors.

  • Most pairs of 2D vectors span all 2D space.

  • If the two vectors align, their span is restricted to a single line.

  • The span of zero vectors is just the origin.

Conceptualizing Vectors as Points

• Vectors can be visualized as points when dealing with collections:

  • Individual vectors are arrows with tails at the origin.

  • Collections represented by points at the tips of vectors.

• For pairs of vectors, their span is conceptualized as the entire 2D plane, except in special cases (aligned or zero vectors).

Extending to Three Dimensions

• In 3D space:

  • The span of two non-aligned vectors is a flat sheet through the origin.

  • Adding a third vector can change the span:

    • If aligned with the first two, the span remains the same.

    • If not aligned, it spans the entire 3D space.

Linear Independence and Dependence

• Linearly Dependent: If one vector can be expressed as a combination of others, resulting in no new dimensions.

• Linearly Independent: All vectors contribute new dimensions to the span, none can be expressed as a combination of the others.

Basis of a Space

• A basis of a space is a set of linearly independent vectors that span the space.

• This definition aligns with vectors that provide full dimensionality without redundancy.