Vectors | Chapter 1, Essence of linear algebra

About this video

• Video Title: Vectors | Chapter 1, Essence of linear algebra
• Channel: 3Blue1Brown
• Speakers: Grant Sanderson
• Duration: 00:09:52

Overview

This video introduces the fundamental concept of vectors in linear algebra, explaining three different perspectives: the physics student's view of arrows in space, the computer science student's view of ordered lists of numbers, and the mathematician's generalized approach. It emphasizes the importance of vector addition and scalar multiplication as core operations and explains how to translate between the geometric (arrow) and numerical (list) representations of vectors.

Key takeaways

• Definition of a Vector: Vectors can be understood as arrows defined by their length and direction (physics), or as ordered lists of numbers (computer science). Mathematicians generalize this to any entity with sensible addition and scalar multiplication.
• Coordinate System: Vectors in linear algebra are typically visualized with their tails at the origin of a coordinate system (e.g., x, y, z axes).
• Numerical Representation: In 2D, a vector is represented by a pair of numbers (x, y) indicating movement along the axes. In 3D, it's a triplet (x, y, z).
• Vector Addition: Geometrically, vector addition is achieved by placing the tail of the second vector at the tip of the first. Numerically, it involves adding corresponding components of the vectors.
• Scalar Multiplication: Multiplying a vector by a scalar (a number) scales its length (stretching or squishing) and can reverse its direction if the scalar is negative. Numerically, this means multiplying each component of the vector by the scalar.
• Translational Utility: The power of linear algebra lies in the ability to translate concepts between the geometric and numerical representations of vectors, which is useful in fields like data analysis and computer graphics.