#9 and #10 1 2 4 9, let T : R3 ? R3 be defined as T(E-10 2-2! (a) Determine if T
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#9 and #10
1 2 4 9, let T : R3 ? R3 be defined as T(E-10 2-2! (a) Determine if T is one-to-one. If T is not one-to-one, then give two vectors 21,#2 E R3 such that (b) Determine if T is onto. If T is not onto, then give a vector not exist i satisfying T(2) = e R3 such that there does 10. Let A be an n x n matrix. (a) Suppose A is invertible. Prove that A and A-1 have the same eigenvectors. How are the eigenvalues of A1 relate to the eigenvalues of A? (b) Prove that A is not invertible if and only if 0 is an eigenvalue of A.Explanation / Answer
9. (a).Let the standard matrix of T be denoted by A. The RREF of A is
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It shows that the 3rd column of A is a linear combination of its first 2 columns so that the columns of A are not linearly independent. Hence T is not one-to-one.Let X1=(x,y,z)T and X2=(a,b,c)T be 2 vectors such that T(X1)=T(X2).Then AX1=AX2 so that (x+2y-4z,2y-2z,x-y-z)T = (a+2b-4c, 3b-2c,a-b-c)T. Then x+2y-4z = a+2b-4c, 2y-2z = 2b-2c and x-y-z = a-b-c. Now, if x = 3,y = 2 ,z = 1 , a = 5,b=3,c = 2, then all these conditions are satisfied. Thus, T(3,2,1)T = T(5,3,2)T.
(b).It is apparent from the RREF of A that the columns of A do not span R3. Hence T is not onto. The vector (x,y,z)T where z? 0 is not the image under T of any vector in R3.
Please post Q. 10 again separately.
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