From: 3blue1brown
In linear algebra, vector coordinates provide a fundamental way to describe vectors numerically, creating a bridge between pairs of numbers (or tuples) and multi-dimensional vectors [00:00:16]. A deeper understanding of these coordinates is central to the field [00:00:30].
Coordinates as Scalars
When a pair of numbers, like (3, -2), describes a vector, each coordinate should be thought of as a scalar [00:00:36]. These scalars indicate how much to stretch or squish other vectors [00:00:40].
In the standard XY coordinate system, there are two special vectors:
- i-hat: The unit vector pointing right along the x-axis [00:00:48].
- j-hat: The unit vector pointing straight up along the y-axis [00:00:52].
The x-coordinate of a vector scales i-hat, and the y-coordinate scales j-hat [00:01:02]. The described vector is the sum of these two scaled vectors [00:01:14].
Basis of a Coordinate System
The vectors i-hat and j-hat are collectively called the basis of a coordinate system [00:01:27]. This means that when coordinates are viewed as scalars, the basis vectors are what these scalars actually scale [00:01:34].
Importance of Alternate Coordinate Systems
It’s possible to choose different basis vectors to create a new, valid coordinate system [00:01:54]. For example, two non-aligned vectors can serve as new basis vectors [00:02:01]. By scaling these new basis vectors and adding them, any two-dimensional vector can be reached [00:02:17].
This highlights a crucial point: whenever vectors are described numerically, it relies on an implicit choice of what basis vectors are being used [00:02:52].
Linear Combination
The process of scaling two or more vectors and adding them together is called a linear combination [00:03:02]. One way to conceptualize the “linear” aspect is that if one scalar is fixed and the other varies, the tip of the resulting vector traces a straight line [00:03:18].
Span of Vectors
The span of a given pair of vectors is the set of all possible vectors that can be reached by a linear combination of those vectors [00:04:02]. Essentially, it asks what vectors can be reached using only vector addition and scalar multiplication [00:04:31].
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In 2D Space:
- For most pairs of two-dimensional vectors, their span is all vectors of 2D space (the entire plane) [00:03:36].
- If the two original vectors are aligned (point in the same or opposite direction), their span is limited to a single line passing through the origin [00:03:43].
- If both vectors are zero, the span is just the origin [00:03:53].
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In 3D Space:
- If two vectors in 3D space are not pointing in the same direction, their span is a flat sheet (a plane) cutting through the origin [00:06:04].
- Adding a third vector:
- If the third vector is already on the span of the first two, the span does not change [00:07:26].
- If the third vector is not on the span of the first two, it unlocks access to every possible three-dimensional vector, sweeping the plane through all of space [00:07:42].
Visualizing Collections of Vectors
When dealing with collections of vectors, it’s common to represent each one with just a point in space, specifically the point at the tip of the vector when its tail is at the origin [00:04:47]. This simplifies visualization:
- A collection of vectors sitting on a line can be thought of as the line itself [00:05:10].
- All two-dimensional vectors can be conceptualized as the infinite flat sheet of two-dimensional space [00:05:19].
Linear Dependence and Independence
This terminology describes whether vectors add new dimensions to a span:
- Linearly Dependent: A set of multiple vectors where at least one vector is redundant, meaning its removal does not reduce the span [00:08:23]. This implies one vector can be expressed as a linear combination of the others [00:08:40].
- Linearly Independent: A set of vectors where each vector genuinely adds another dimension to the span [00:08:52].
The technical definition of a basis of a space is a set of linearly independent vectors that span that space [00:09:12].