NameMath 311 Exam 3 Spring 2010Section 502 P. YasskinConsider the vector space P3of polynomialswith degree less than 3 with standard basise1= 1 e2= x e3= x2and the linear operatorL : P3 P3: L(p)=(x − 2)dpdx1,2 / 4 10 /153,4 /12 11,12 /125 /10 13 /156,7 /12 14,15 /108,9 /16 Total /1061.(2 pts) What is dimP3? What is the size of the matrix A of the linear map L?dimP3= A is a matrix.2.(2 pts) Let p = a + bx + cx2. Compute L(p).L(p)=3.(10 pts) Identify the kernel of L, a basis for the kernel, and the dimension of the kernel.dimKer(L)=4.(2 pts) What does the kernel of L, (found in part 2), say about one of the eigenvalues of L,and the eigenpolynomial(s) for that eigenvalue? No new computations!5.(10 pts) Identify the image of L, a basis for the image, and the dimension of the image.dimIm(L)=16.(6 pts) Is the function L one-to-one? Why?7.(6 pts) Is the function L onto? Why?8.(6 pts) Find the matrix of the linear map L relative to the e basis. Call ite←eA .e←eA =9.(10 pts) Find the eigenvalues ofe←eA .λ1= λ2= λ3=210.(15 pts) Find the eigenvectors for each eigenvalue. Call them v1, v2and v3.a.λ1= :v1=b.λ2= :v2=3c.λ3= :v3=11.(6 pts) Find the eigenpolynomials for each eigenvalue. Call them q1, q2and q3.Verify they are eigenpolynomials by checking that L(qk)= λkqkusing the definition of L.q1= L(q1)=q2= L(q2)=q3= L(q3)=412.(6 pts) The eigenpolynomials q=(q1,q2,q3)form a second basis for P3.Find the matrix of the linear map L relative to the q basis. Call itq←qD .q←qD =13.(15 pts) Find the change of basis matriceseqC andqeC .eqC =qeC =514.(5 pts) Verify your matrices satisfyeqCq←qDqeC=e←eA .15.(5 pts)
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