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anonymous
 5 years ago
Let P₃(R) be the vector space of polynomials over R of degree strictly less than 3. Let T: P₃ (R) →P₃ (R) be defined by T(⨍)= 2⨍’
(a) Show that T is a linear transformation
(b) Find a basis for the kernel of T
(c) Is T an isomorphism? Justify your answer.
anonymous
 5 years ago
Let P₃(R) be the vector space of polynomials over R of degree strictly less than 3. Let T: P₃ (R) →P₃ (R) be defined by T(⨍)= 2⨍’ (a) Show that T is a linear transformation (b) Find a basis for the kernel of T (c) Is T an isomorphism? Justify your answer.

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watchmath
 5 years ago
Best ResponseYou've already chosen the best response.1\(T(f+g)=2(f+g)'=2(f'+g')=2f'+2g'=T(f)+T(g)\) \(T(\alpha f)=2(\alpha f)'=2\alpha f'=\alpha(2f')=\alpha T(f)\) Hence \(T\) is linear. \(f\in \ker T\iff T(f)=2f'=0\iff f'=0\iff f\equiv \text{costant}\) Obviously \(T\) is not an isomorphims since it is not injective ( because \(\ker T\neq 0\) )
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