q4rs - is S ⃗x B = ⃗x E and that you are looking for...

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Quiz IV 1. The vector ⃗x = (5 , 6 , 9) lies in the plane P spanned by ⃗v 1 = (1 , 0 , - 1) and ⃗v 2 = (5 , 3 , 2). Find the coordinates of ⃗x with respect to this basis for P . Solution. You need to find the scalars c 1 and c 2 such that 5 6 9 = c 1 1 0 - 1 + c 2 5 3 2 You can either do that by row reduction, or by noticing that the 0 in ⃗v 1 makes c 2 = 2 and from that c 1 = - 5. 2. Find the matrix of the transformation T ( ⃗x ) = A⃗x , where A = [ 3 4 - 1 - 1 ] with respect to the basis ⃗v 1 = (2 , - 1), ⃗v 2 = ( - 5 , 3). Solution: The most important things to remember are that the matrix S = [ 2 - 5 - 1 3 ] 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 MS . In this case that is [ 3 5 1 2 ][ 3 4-1-1 ][ 2-5-1 3 ] = [ 3 5 1 2 ][ 2-3-1 2 ] = [ 1 1 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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