Linear combinations, span, and basis vectors | Chapter 2, Essence of linear algebra
3Blue1Brown・1 minute 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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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.
