We have said in the introduction of this course, that linear algebra has applications in many...

 

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We have said in the introduction of this course, that linear algebra has applications in many field of mathematics. Here is an example of application to differential equations: Le T be the operator in Pn (R) defined by the following property: T(P) = XP" + (X + 2)P' T. a) Prove that T is actually an operator of this vector space. b) Prove that the matrix of T in the standard basis is triangular, and use this to find the eigen values of c) For the following differential equations, decide wether they have a (non zero) polynomial solution, or not. If they have one, what is the dimension of the subspace of polynomial solutions? xf" (x)+(x+2) f'(x) = 3f(x) xf" (x)+(x+2) f'(x) = 4f(x) We have said in the introduction of this course, that linear algebra has applications in many field of mathematics. Here is an example of application to differential equations: Le T be the operator in Pn (R) defined by the following property: T(P) = XP" + (X + 2)P' T. a) Prove that T is actually an operator of this vector space. b) Prove that the matrix of T in the standard basis is triangular, and use this to find the eigen values of c) For the following differential equations, decide wether they have a (non zero) polynomial solution, or not. If they have one, what is the dimension of the subspace of polynomial solutions? xf" (x)+(x+2) f'(x) = 3f(x) xf" (x)+(x+2) f'(x) = 4f(x)