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# Coordinates with respect to a basis

Understanding alternate coordinate systems. Created by Sal Khan.

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• at when you move in the v2 direction aren't you also moving in the v1 direction? • 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. • 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? • where did the essence of linear algebra go • 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".
• Are a vector's coordinates with respect to a basis always unique? If yes, why? • 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!
• At why did he say that the basis B had K vectors? wouldn't it have n vectors because the vector space V is n dimensional? isn't the number of terms of a basis equal to the dimension of the space it's a basis for?
EDIT: Oh! Just because the vector space V is in R^n, doesn't mean the vector space necessarily encompasses everything in R^n! V could be a giant plane in a 3 dimensional space or a 6-dimensional space-volume-thing in an 8-dimensional space! It could be a line in an x y coordinate system! That's what subspace means! • I am facing difficulty understanding the vecotr component in the 2 different co ordinate systems. Kindly help me.

Suppose I have a vector A on the 2d co ordinate system x and y and therefore vector A = Axi + Ayj where i and j are unit vectors along x and y axis.

Suppose a new co ordinate system is introduced in which is x'y' and is titlted from the xy plane by angle theta and both the co ordinate systems (xy and x'y') share the same origin.

The component of the vector seem to change on x'y' plane.

A = i'Ax' + j'Ay' (i' and j' are the unit vectors along the x' and y')

and now i' = i cos (theta) + j sin (theta)
j' = - i sin (theta) + j cos (theta)

Kindly help me understand the concept.
(1 vote) • The actual positions of the vectors haven't changed, only the way they are represented. In the first co-ordinate system they are the same, in the second they have different values, because the coordinate system doesn't overlay the first. It's quite useful in physics, when certain coordinate systems yield much simpler math to solve a problem. Even if the answer needs to be in the first coordinate system, once solved it can easily be translated back. 