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# Coordinates with respect to a basis

Understanding alternate coordinate systems. Created by Sal Khan.

## Want to join the conversation?

• at when you move in the v2 direction aren't you also moving in the v1 direction?
• Technically, yes. You're partially moving in the same direction of both vectors.That's why the graph paper is skewed. If the vectors were orthogonal you would move according to the standard basis / normal, unskewed graph paper.
• If i know the global coordinates of a point , How can I find it in my local coordinates. Say the global coordinates of my point is x1,y1,z1 . And my new coordinate system has three basis vectors, e1( e1x,e1y,e1z),e2( e2x,e2y,e2z),e3( e3x,e3y,e3z). How will find the coordintes in my new coordinate system.
• Refer to the following video, "Change of Basis Matrix."
• When we change from a basis to another, does the origin (the intersection of our reference "axes") remain in the same point? In other words, does the origin always have coordinates equal to the zero vector, no matter what basis we consider?
• Yes, because c's for it are always 0, 0, ... 0, no matter what the basis is.
• I was just wondering if back in fourth grade when I was learning how to graph coordinates, why they didn't just tech me this even though it is harder it would have been easier just to have learnt everything all about graphs from the very start, so that later in life I would already know how to do it. And the coordinate plans are easy to understand but. I was just wondering how to exactly make the coordinate plane easier to draw and write in the numbers? Any ideas?
• Imho they cannot teach you everything so there must be a point where they stop. And that point just happens to be an R^2 plane and everything about R^2. Take into consideration fact that Sal shows this in R^2 for easier understanding, but it's more of a R^n connected thing. So it's more of linear algebra than basic algebra and you cannot teach fourth graders whole linear algebra :) All in all, it is not connected with rest of school algebra that much so it isn't necessary there.
For the second part of the question I don't understand it though :) What do you mean by "plane easier to draw".
• where did the essence of linear algebra go
• The essence of linear algebra is spectacular series of videos created by YouTube user 3blue1brown and you can watch them here:
https://youtu.be/kjBOesZCoqc?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab
• Are a vector's coordinates with respect to a basis always unique? If yes, why?
• They are unique to that particular basis. The reason is because two vectors are equal by definition if and only if their coordinates are equal (and this is true regardless of basis), so if a vector had two coordinate representations in the same basis, those two have to be the same, otherwise we would contradict what it means for a vector to equal itself.
• I'm just curious if we could do examples of finding the coordinate vectors, transition matrices, from Polynomial spaces or even Matrice spaces (i.e. P2(t) and M2x2). I understand the logic I think, but the only examples I can find are always using vectors, but my hw uses Polynomial spaces and matrix spaces. So I'm finding it hard to transition this logic from one to the other.. Thank you :)
• Try and think of it as a together type thing and that matrices spaces and polynomial spaces are related in their expressions in logic. Their formulas are different but the logic in polynomial spaces tells us that the examples show us that it is easier to understand when you do the matrices spaces also, learning at the same time. I found it easier, hope this helps!