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Lecture on Vector Coordinates, Basis and Span
May 31, 2024
Lecture on Vector Coordinates, Basis and Span
Introduction
Review on vector addition and scalar multiplication
Discussion of vector coordinates in the context of linear algebra
Vector Coordinates
Pairs of numbers signify vectors (e.g., (3, -2))
Coordinates as scalars that stretch/squish vectors
Basis Vectors
Special vectors in the x-y system:
i-hat
: unit vector in the x direction (1, 0)
j-hat
: unit vector in the y direction (0, 1)
Coordinates can scale these basis vectors:
x-coordinate scales i-hat
y-coordinate scales j-hat
Vector described as a sum of scaled basis vectors
Basis
: The set of vectors that the coordinates scale
Alternative Basis
New coordinate systems can be generated with different basis vectors
By choosing different basis vectors, the numerical description of vectors changes
Challenge to think about vectors generated by different pairs of basis vectors
Linear Combinations
Adding scaled vectors from chosen scalars
The term 'linear' refers to the resulting vector drawing a straight line when one scalar is fixed
Linear combination of two vectors can:
Reach every possible 2D vector (if vectors don’t line up)
Be limited to a line (if vectors line up)
Be at origin (if both vectors are zero)
Span
: The set of all possible vectors from linear combinations of a given pair of vectors
Vector Points
Vectors often represented as points for simplicity
Crowding avoided by representing collections of vectors as points
Single vectors = arrows; collections of vectors = points
Span in 3D Space
Two vectors in 3D space span a flat sheet (plane) through the origin
Adding a third vector to span:
If third vector within the span of first two, no new vectors reached
Generally, a third vector unlocks access to full 3D space
Linear Dependence and Independence
Linearly dependent
: A vector can be expressed as a linear combination of others
Linearly independent
: Each vector adds a new dimension to the span
Puzzle and Conclusion
Technical definition of basis: Set of linearly independent vectors that span a space
Challenge to think about why this definition makes sense
Next topic preview: Matrices and transforming space
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