Vectors | Chapter 1, Essence of linear algebra
Introduction to Vectors
This video introduces the concept of vectors and their different perspectives. It explains how vectors can be viewed as arrows in space, ordered lists of numbers, or abstract mathematical objects.
Three Perspectives on Vectors
- There are three distinct but related ideas about vectors: the physics student perspective, the computer science student perspective, and the mathematician's perspective.
- The physics student perspective views vectors as arrows pointing in space with length and direction defining a given vector.
- The computer science student perspective views vectors as ordered lists of numbers where order matters and each number represents a feature or dimension.
- The mathematician's perspective generalizes both views by saying that a vector can be anything where there is a sensible notion of adding two vectors and multiplying a vector by a number.
Geometric View of Vectors
- Whenever introducing a new topic involving vectors, think about an arrow inside a coordinate system with its tail sitting at the origin.
- In linear algebra, it's almost always the case that your vector will be rooted at the origin.
- Once you understand a new concept in the context of arrows in space, translate it over to the list-of-numbers point-of-view by considering the coordinates of the vector.
Coordinate System for Two-Dimensional Space
- A coordinate system for two-dimensional space has a horizontal line called x-axis and vertical line called y-axis intersecting at origin.
- Tick-marks on each axis represent an arbitrary length to represent 1.
- Coordinates of a vector is represented by an ordered pair of numbers giving instructions for how to get from the tail of that vector at the origin to its tip.
Coordinate System for Three-Dimensional Space
- In three dimensions, you add a third axis called z-axis perpendicular to both x and y axes.
- Each vector is associated with an ordered triplet of numbers where the first tells you how far to move along the x-axis, the second tells you how far to move along the y-axis, and the third tells you how far to move along the z-axis.
Introduction to Linear Algebra
In this section, the instructor introduces the two fundamental operations in linear algebra, vector addition and scalar multiplication. He explains how these operations are defined and why they are reasonable.
Vector Addition
- To add two vectors, move the second one so that its tail sits at the tip of the first one; then draw a new vector from the tail of the first one to where the tip of the second one now sits.
- This definition of addition is reasonable because each vector represents a certain movement in space. Taking a step along the first vector and then taking a step in the direction and distance described by the second vector is equivalent to moving along their sum.
- Numerically, adding two vectors involves matching up their terms and adding each one together.
Scalar Multiplication
- Multiplying a given vector by a scalar means stretching or squishing it by that factor. Scaling is when numbers act like scalars on vectors.
- Numerically, multiplying a given vector by a scalar means multiplying each component by that scalar.
The Usefulness of Linear Algebra
In this section, the instructor discusses why linear algebra is useful for data analysis, physics, computer graphics programming, and more.
- Linear algebra provides a way to conceptualize many lists of numbers visually, which can clarify patterns in data and give a global view of what certain operations do.
- Linear algebra also provides physicists and computer graphics programmers with a language to describe space and the manipulation of space using numbers that can be crunched and run through a computer.