Change of basis | Essence of linear algebra, chapter 9
How do you translate back and forth between coordinate systems that use different basis vectors? Full series: http://3b1b.co/eola Future series like this are ...
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If I have a vector sitting here in 2D space we have a standard way to describe it with coordinates. In this case, the vector has coordinates [3, 2], which means going from its tail to its tip involves moving 3 units to the right and 2 units up. Now, the more linear-algebra-oriented way to describe coordinates is to think of each of these numbers as a scalar a thing that stretches or squishes vectors. You think of that first coordinate as scaling i-hat the vector with length 1, pointing to the right while the second coordinate scales j-hat the vector with length 1, pointing straight up. The tip to tail sum of those two scaled vectors is what the coordinates are meant to describe. You can think of these two special vectors as encapsulating all of the implicit assumptions of our coordi...
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If I have a vector sitting here in 2D space we have a standard way to describe it with coordinates. In this case, the vector has coordinates [3, 2], which means going from its tail to its tip involves moving 3 units to the right and 2 units up. Now, the more linear-algebra-oriented way to describe coordinates is to think of each of these numbers as a scalar a thing that stretches or squishes vectors. You think of that first coordinate as scaling i-hat the vector with length 1, pointing to the right while the second coordinate scales j-hat the vector with length 1, pointing straight up. The tip to tail sum of those two scaled vectors is what the coordinates are meant to describe. You can think of these two special vectors as encapsulating all of the implicit assumptions of our coordi...
