LoginGet started

Linear combinations, span, and basis vectors | Chapter 2, Essence of linear algebra

3Blue1Brown・2 minutes read

Vector coordinates are pairs of numbers representing two-dimensional vectors, with each coordinate acting as a scalar to manipulate vectors. Basis vectors i-hat and j-hat are scaled by the x and y coordinates, respectively, in the xy coordinate system, affecting vector descriptions and the span of scaled vectors.

Insights

  • The xy coordinate system utilizes basis vectors i-hat and j-hat to scale vectors in the x and y directions, respectively, influencing the overall vector sum through their scalar multiplication.
  • Varying basis vectors in a coordinate system alter how vectors are described, showcasing the importance of understanding the impact of different bases on vector representations and linear combinations.

Get key ideas from YouTube videos. It’s free

Recent questions

  • What are vector coordinates?

    Pairs of numbers representing two-dimensional vectors.

Related videos

Summary

00:00

Understanding Vector Coordinates and Basis Vectors

  • Vector coordinates involve pairs of numbers representing two-dimensional vectors.
  • Each coordinate can be seen as a scalar that stretches or squishes vectors.
  • The xy coordinate system has special vectors: i-hat (right) and j-hat (up).
  • X coordinate scales i-hat, y coordinate scales j-hat, resulting in a sum of scaled vectors.
  • Basis vectors i-hat and j-hat form the basis of a coordinate system.
  • Different basis vectors lead to varied coordinate systems, affecting vector descriptions.
  • Linear combinations of scaled vectors involve adding two vectors, forming the span of those vectors.
Channel avatarChannel avatarChannel avatarChannel avatarChannel avatar

Try it yourself — It’s free.