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Example using orthogonal change-of-basis matrix to find transformation matrix

Example using orthogonal change-of-basis matrix to find transformation matrix. Created by Sal Khan.

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• at , i dont understand why T(v3) is -v3. Since it is perpendicular to the plane spanned by v1 and v2..shouldnt the projection of v3 on the plane be 0?
• He mixed projection with reflection. If it were the case for reflection, then T(v3) has to be equal to -v3 which he mentions at .
• Can we always consider that v1 equals v1, v2 equals v2 and v3 equals -v3 or is it just a convention of this example?
• How v1, v2, and v3 change describes the result of a transformation. In this case, v3 => -v3, which means that a transformed vector will be flipped around the plane formed by v1 and v2. If Sal used v3 => 2 * v3, then the transformation would double the distance between a vector and a plane formed by v1 and v2.
• At , why is it that v1 = 1*v1 + 0*v2 + 0*v3 as a vector in the new basis B={v1,v2,v3}?
• All vectors in that R^3 can be represented as linear combinations of v1, v2, and v3. That is, all vectors in R^3 can be expressed as x = c1v1 + c2v2 + c3v3, where c1, c2, and c3 are scaling factors. If x = v1, what are the values for c1, c2, and c3?
• Around he is doing the transformations of the vectors. What exactly are the transformations of? The mirror image perpendicular to the plane?
(1 vote)
• Yes, it is the mirror image of the vector through the plane.
• In the earlier videos we established that if C is the change of basis matrix, Xb is a vector X with respect to the basis B and X is a vector with respect to the standard coordinates (our basis), then C * Xb = X.
inv(C) is then our basis' coordinates in basis B's coordinate system. Thus, inv(C) * X = Xb

What I can't understand here is how we have assumed inv(C) to be our basis ( v1, v2, v3's) coordinates with respect to the plane's coordinate system (or plane's basis)? Why does inv(C) * X give us a vector in the plane's basis and when exactly did we establish that?
(1 vote)
• We established that at the beginning when drawing the 3 orthonormal vectors (2 orthogonal ones on the plane and a third blue one orthogonal to both).

Also, inv(C) isn't our basis. (C) is our basis.
(1 vote)
• In order for him to be able to deduce that Inverse(C) = Transpose(C), the matrix property A * Inverse(A) = Inverse(A) * A = I, must hold, right?
(1 vote)
• At Sal takes for granted that [-v3]_B is the same as -[v3]_B.
But have we proved that?
Does a scaled up vector in one basis S always scale up by the same amount in another basis B?
(1 vote)
• I think sal made a mistake in the final multiplication