Question: Let B = (U1, U2, U3) be a basis for R3. Given a vector w in R3, we have defined the coordinates of w with

Let B = (U1, U2, U3) be a basis for R3. Given a vector w in R3, we have defined the coordinates of w with respect to B to be 91 [w] B = 92 93 where q1, 92, and 93 are the unique scalars which satisfy the vector equation w = q101 + 9202 + 9303. a. Show that for every vector w, we have [w]B = C-'w, where C is the change of basis matrix associated to B. Hint: Rewrite the vector equation which defines q1, 92, 93 as a matrix equation. b. If A is any 3 x 3 matrix and w is any 3D vector, show that [AwlB = MB(A) [w]B. In other words, the matrix MB(A) tells us how the B-coordinates of an arbitrary vector get trans- formed, when we multiply that vector by A. Hint: First show that C-1A = MB(A)C-1, using the definition of MB( A)

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