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.

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