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.
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.
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 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.
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.
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.
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).
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.
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.
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.