Canonical form of t will contain one jordan block of

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canonical form of T will contain one Jordan block of length 1 corresponding to λ 2 = 6. To find a basis for K 6 = E 6 consisting of cycles, it suffices to find any 6-eigenvector of T; working in β 0 -coordinates and solving the matrix equation ( B - 6 I ) x = 0, we see that we may take β 2 = 1 - 3 3 0
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as our basis for K 6 . Thus, the Jordan canonical form of T is J = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 , and a Jordan canonical basis is β = β 1 β 2 = 1 0 0 0 , 0 1 1 0 , 0 0 0 1 , 1 - 3 3 0 . (f) V is the vector space of polynomial functions in two real variables x and y of degree at most 2 , as defined in Example 4, and T is the linear operator on V defined by T ( f ( x, y ) ) = ∂x f ( x, y ) + ∂y f ( x, y ) .
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dots. Thus, the dot diagram for λ = 0 must be (T - 0 I) 2 v (T - 0 I) w u (T - 0 I) v w v Proceeding as in part (b), we first find v such that T 2 v 6 = 0; we may take v = x 2 , which gives the cycle β 1 = (T - 0 I) 2 v , (T - 0 I) v , v = 2 , 2 x, x 2 . Next, to find a cycle { (T - 0 I) w , w } of length two, we find w such that (T - 0 I) 2 w = 0 but (T - 0 I) w 6 = 0, taking care to ensure that the resulting cycle is linearly independent of the previously constructed cycle. One might be tempted to guess w = y , which will generate a cycle of length 2; however, the initial vector of the resulting cycle would be 1, which is not linearly independent of the previous cycle. However, taking w = x 2 - y 2 yields the cycle β 2 = (T - 0 I) w , w = 2 x - 2 y , x 2 - y 2 , which does the job. Finally, to get our last cycle of length 1 we just need a 0-eigenvector of T that is linearly independent of the previously collected vectors. Examining the matrix A (or using Math 54 techniques to find its nullspace), we see that we can take β 3 = { u } = { 2 xy - x 2 - y 2 } . Putting all of this together, we obtain the Jordan canonical form J = 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 and a Jordan canonical basis β = β 1 β 2 β 3 = { 2 , 2 x, x 2 , 2 x - 2 y, x 2 - y 2 , 2 xy - x 2 - y 2 } . 9

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