q4ts - is S ⃗x B = ⃗x E and that you are looking for...

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Quiz IV 1. The vector ⃗x = (4 , 7 , 13) lies in the plane P spanned by ⃗v 1 = (1 , 1 , 1) and ⃗v 2 = (2 , 3 , 5). Find the coordinates of ⃗x with respect to this basis. Solution. You need to find the scalars c 1 and c 2 such that 3 7 13 = c 1 1 1 1 + c 2 2 3 5 You should probably do that by row reduction: 1 2 | 4 1 3 | 7 1 5 | 13 1 2 | 4 0 1 | 3 0 3 | 9 1 0 | - 2 0 1 | 3 0 0 | 0 So c 1 = - 2 and c 2 = 3. 2. Find the matrix of the transformation T ( ⃗x ) = A⃗x , where A = [ 5 2 - 4 - 1 ] , with respect to the basis ⃗v 1 = (1 , - 1), ⃗v 2 = ( - 1 , 2). Solution: The most important things to remember are that the matrix S = [ 1 - 1 - 1 2 ] will change coordinates in the new basis into coordinates in the standard basis, that
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Unformatted text preview: is S [ ⃗x ] B = [ ⃗x ] E , and that you are looking for the matrix M such that M [ ⃗x ] B = [ T ( ⃗x )] B . Since A [ ⃗x ] E = [ T ( ⃗x )] E , substituting gets you AS [ ⃗x ] B = S [ T ( ⃗x )] B , and M = S-1 AS . In this case S-1 AS is [ 2 1 1 1 ][ 5 2-4-1 ][ 1-1-1 2 ] = [ 2 1 1 1 ][ 3-1-3 2 ] = [ 3 1 ] . You could also do this by the one of the other methods discussed in Bretscher, but, when the inverse of S is easy to find, the method here is easy....
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