i. What is the matrix of T in the standard ordered basis for R2 ?
ii. What is the matrix of T in the ordered basis B = {α1, α2}, where α1 = (1, 2) and α2 = (1, −1)?
iii. Prove that for every real number c the operator (T − cI) is invertible.
1
Expert's answer
2021-01-08T14:39:27-0500
Here, we have
T(x,y)=(−y,x)=[−yx]
(i) Standard ordered basis for R2is[1001]
So, for the first column applying the linear operator T we have [01] and for the second column after applying the linear operator T we have [−10] . So, the matrix for T is A(let):-
A=[01−10]
(ii) New basis is:-
B=[121−1]
Now, we have to find the matrix A wrt the basis B.
Now we take the first column of B and left multiply with A.
A[12]=[01−10]⋅[12]=[−21]
Now, solving for a, b where
[−21]=a[12]+b[1−1]⇒[−21]=[a+b2a−b]
Solving we get, a=3−1 and b=3−5 .
Similarly we do the above steps with the second column of B and find c, d:-
[11]=[c+d2c−d]
Solving, we have c=32 and d=31 .
Therefore the final matrix wrt, the given basis B is:-
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