Definition of Vectors: Vectors are fundamental building blocks in linear algebra, defined from three perspectives:
Physics Perspective: Vectors are arrows in space characterized by their length and direction. They can be moved around as long as these two properties remain unchanged.
Computer Science Perspective: Vectors are ordered lists of numbers, where the order is significant. For example, a house can be represented as a two-dimensional vector with square footage and price.
Mathematical Perspective: Vectors can be generalized to any entity where vector addition and scalar multiplication are defined, although this view is more abstract.
2D Coordinate System: In a two-dimensional space, vectors are represented as pairs of numbers (coordinates) that indicate movement along the x-axis and y-axis from the origin.
3D Coordinate System: In three-dimensional space, vectors are represented as triplets of numbers, adding a z-axis perpendicular to the x and y axes.
Vector Addition: To add two vectors, the tail of the second vector is placed at the tip of the first, and a new vector is drawn from the tail of the first to the tip of the second. This operation reflects the idea of combining movements in space.
Scalar Multiplication: Multiplying a vector by a scalar stretches or shrinks the vector. For example, multiplying by 2 doubles its length, while multiplying by -1 flips its direction.
The operations of vector addition and scalar multiplication are central to linear algebra. Understanding these operations allows for the translation between geometric and numerical representations of vectors.
The ability to switch between these perspectives enhances data analysis and visualization, providing clarity in understanding patterns and operations.
Vector coordinates can be thought of as scalars that stretch or squish vectors in the coordinate system.
The unit vectors i-hat and j-hat are fundamental, representing the x and y directions respectively, and are used to scale vectors based on their coordinates.
The vectors i-hat and j-hat together form the basis of a coordinate system, allowing any vector to be expressed as a linear combination of these basis vectors.
Different pairs of basis vectors can be chosen, leading to different coordinate systems while still representing all possible two-dimensional vectors.
A linear combination of vectors involves scaling and adding them, which can be visualized as drawing a straight line when one scalar is fixed and the other varies.
The span of two vectors is the set of all possible vectors that can be formed through linear combinations of those vectors, which can fill the entire two-dimensional space unless the vectors are collinear.
Vectors can be represented as points in space, where the tip of the vector indicates its position, simplifying the visualization of spans and collections of vectors.
In three-dimensional space, the span of two non-collinear vectors forms a flat sheet, while adding a third vector can either expand the span to all of 3D space or remain confined to the existing span if it is dependent on the first two.
Linear dependence occurs when one vector can be expressed as a combination of others, while linear independence means each vector contributes a new dimension to the span.
A basis of a space is defined as a set of linearly independent vectors that span that space, which aligns with the concepts of span and linear independence discussed.