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# Coordinates with respect to orthonormal bases

Seeing that orthonormal bases make for good coordinate systems. Created by Sal Khan.

## Want to join the conversation?

• Hi,

what does he mean by 'B coordinate representation of x '? I do not understand what he means by 'coordinate' here. Does he meant the projection of x on the span of V? thanks.
• Here's a link to a video on the topic which begins a 7 part series of videos explaining it in detail.
Also, if you like, I have summarized a bit of it here:
The standard basis is {e_1,e_2,...,e_n} for R^n.
If we have a k-dimensional subspace V of R^n, say it has a basis B = {v_1,v_2,...,v_k}, where the v_i's are in R^n.
We could write standard coordinates for a vector x in V as
x = <x_1,x_2,...,x_n> = x_1*e_1 + x_2*e_2 + ... + x_n*e_n.
But we could also represent x as a linear combination of the v_i's. We write this representation of x as [x]_B, and this is the B coordinate representation of the vector x.
If x = c_1*v_1 + c_2*v_2 + ... + c_k*v_k, then we write
[x]_B = <c_1,c_2,...c_k>_B. We call these c_i's the B coordinates of x.
Hope that summary helps, but I do suggest watching the videos linked above. They contain some minor mistakes, but if you read the comments and push through them you should be better set up for what continues here.
• Is it a side effect of the properties discussed in this video that A^-1 = A^t if A's columns form an orthonormal set?
• I had noticed it too. It turns out that such matrices have a special name: Orthogonal matrix.
(1 vote)
• How could you describe the takeaway here in general terms?

I came away with: "It is much easier to change bases this way. Just dot your vectors of an orthonormal set with a member of R2", but I don't know in what context you would have the vector in R2. Does this question make sense?

We used vector x=(9, -2) in the video, but where would we find this vector otherwise? If we can pick any random vector for x, what can be said of the resulting matrix [x]B?
• Isn't there something wrong with Mr. Khan's transformation? Here's my concern:
I know the coordinates in terms of basis e1, e2, e3 (A = 9, -2). I have another orthonormal basis f1, f2, f3. If I want to know the coordinates in terms of basis f1, f2, f3, wouldn't I do P*A, where P is a rotation matrix defined by: (row 1: e1.f1 e2.f1) (row 2: e1.f2 e2.f2) It seem the method Mr. Khan is using uses a transform/ inverse of this matrix, which is confusing me (. represents dot product)

Source: http://ocw.mit.edu/courses/aeronautics-and-astronautics/16-07-dynamics-fall-2009/lecture-notes/MIT16_07F09_Lec03.pdf
(1 vote)
• Everything is ok, you mixed A and C matrices (transformation and basis change).

In this video: x_b = C^(-1)x, where C^(-1) = transpose of C (in orthonormal case)
C - change of basis matrix, where vectors of basis B are columns in this matrix, so: Cx_b=x

When you are talking about rotation, you mean transformation matrix A.
Relation C and A: A=CDC^(-1), where D is transformation matrix for T with respect do basis B.

When you transform (rotate, scale, shift) a point, you don't change it's basis.
In other words: basis changing doesn't move a point, it makes different reference system, transformation moves a point.
• At he says the Change of Basis Matrix "C" isn't always going to be invertible or square. I thought that, by definition, a Change of Basis Matrix is invertible and square. What am I misunderstanding?
• Suppose you had a Basis B that was linearly independent but it's change of basis matrix C was not invertible, and let's also suppose that x is a member of the subspace that B spans. Couldn't you solve for [x]_sub_B in (C[x]_sub_B = x) by multiplying both sides of the equation by (C Transpose) and the inverse of ( (CTranspose) C). This would make your matrix on the left side of the equation invertible. So our solution should be
[x]_sub_B = [ (C Transpose) C]^(-1) (C Transpose) x.

A basis for a subspace is always linearly independent so if the product of the transpose of the change of basis matrix and the regular change of basis matrix: (C Transpose) C will always be a square matrix composed of linearly independent columns. Right?
(1 vote)
• How to calculate the coordinates on a point
(1 vote)
• Does Salman ever explain Parserval Thereom?
(1 vote)
• what does [x] means ?
(1 vote)
• I think u made a mistake at 13.40, it (-4/5)*(-2) and (3/5)*(-2), or am i wrong
(1 vote)