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

3Blue1Brown・9 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.

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Recent questions

  • What are vector coordinates?

    Pairs of numbers representing two-dimensional vectors.

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Summary

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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.
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