please show work and explanation. thank you Answer the following questions regar
ID: 3185944 • Letter: P
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please show work and explanation. thank you
Answer the following questions regarding the transformation, T, where Tx)-Ax- b and 4 -1 A 2 1 6 -1 8 (a) (5 points) If you know that the line k 1 is a stretch direction for this matrix (i.e, span is an eigenspace for this transformation), what is the stretch factor (cigenvalue) associated with this stretch direction. (b) (5 points) Given that 2 is a stretch factor (eigenvalue) for this matrix. Determine and describe all vectors that have this stretch factor. In other words, describe the set of all stretch directions (eigenvectors) associated with a stretch of 2.Explanation / Answer
(a) Since span{(1,1,1)T} is an eigenbasis for T, hence (1,1,1)T is an eigenvector of A so that A(1,1,1)T = k(1,1,1)T, where k is the eigenvalue of A corresponding to the eigencector (1,1,1)T. Further, A(1,1,1)T = (9,9,9)T = k(1,1,1)T. Hence, k = 9.
(b). Let v= (a,b,c)T be an eigenvector of A corresponding to the eigenvalue 2. Then Av = 2v= (2a,2b,2c)T . Further, Av = A(a,b,c)T = (4a-b+6c,2a+b+6c,2a-b+8c)T. Therefore, 4a-b+6c = 2a or, 2a-b+6c = 0…(1) , 2a+b+6c = 2b or, 2a-b+6c = 0 …(2) and 2a-b+8c = 2c or, 2a-b+6c = 0…(3) Hence b = 2a+6c so that v = (a,2a+6c,c)T = a(1,2,0)T+c(0,6,1)T.
Hence, (1,2,0)T and (0,6,1)T are the 2 eigenvectors of A associated with its eigenvalue 2.
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