Question

In: Math

Let T and S be linear transformations of a vector space V, and TS=ST (a) Show...

Let T and S be linear transformations of a vector space V, and TS=ST

(a) Show that T preserves the generalized eigenspace and eigenspace of S.

(b) Suppose V is a vector space on R and dimV = 4. S has a minimal polynomial of (t-2)² (t-3)²?. What is the jordan canonical form of S.

(c) Show that the characteristic polynomial of T has at most 2 distinct roots and splits completely.

Expert Solution

(a) we have Sx = x for all x in eigenspace E

We want to show that Tx = x

Now TS = ST

=> TSx = STx

=> T(x) = S(Tx)

=> Tx = S(Tx)

Let y = Tx

=> y = Sy

=> y is in eigenspace E

As y=T(x) => Tx is in eigen space E

And hence T preserves the eigenspace of S

(b) Minimal polynomial = (t-2)²(t-3)²

Since dimV = 4

So we have two blocks of 2 and two blocks of 3 in Jordan cannonical form and it is given as :

(c) Since T preserves the generalized eigenspace of S and S has two distinc roots 2 and 3

Hence the characteristic polynomial of T has at most 2 distinc roots

milcah answered 2 years ago

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