Recall that Pn (R) denotes the vector space of poly- nomial of degree at most n...

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Recall that Pn (R) denotes the vector space of poly- nomial of degree at most n with real coefficients in the variable t. We let e₁ = 1,e₂ t, e3 = t², es t³ denote the standard basis in P3. Consider the following functions from P3 (R) to itself. = = = = (1) (Dƒ)(t) f'(t) (=the derivative of the polynomial f). Show that D is a linear transformation and find the matrix A rep- resenting it w.r.t. the standard bases. (2) Show that D is neither injective nor surjective. (3) (Tf) (t) = f(t+1) (for example if f(t) = t² then (Tf)(t) = T(t²) = t² +2t+1). Show that T is a linear transformation and find the matrix At representing it w.r.t. the standard bases. (4) Show that T is both injective and surjective. (5) (E₁ƒ)(t) = ƒ(1)(=the value of f at the point t = 1). Show that E₁ is a linear transformation and find the matrix AE, repre- senting it w.r.t. the standard bases. (6) (Mƒ)(t) = (tf)'. Show that M is a linear transformation and find the matrix AM representing it w.r.t. the standard bases. (7) Are the 4 columns of the representing matrix AM linearly inde- pendent? Recall that Pn (R) denotes the vector space of poly- nomial of degree at most n with real coefficients in the variable t. We let e₁ = 1,e₂ t, e3 = t², es t³ denote the standard basis in P3. Consider the following functions from P3 (R) to itself. = = = = (1) (Dƒ)(t) f'(t) (=the derivative of the polynomial f). Show that D is a linear transformation and find the matrix A rep- resenting it w.r.t. the standard bases. (2) Show that D is neither injective nor surjective. (3) (Tf) (t) = f(t+1) (for example if f(t) = t² then (Tf)(t) = T(t²) = t² +2t+1). Show that T is a linear transformation and find the matrix At representing it w.r.t. the standard bases. (4) Show that T is both injective and surjective. (5) (E₁ƒ)(t) = ƒ(1)(=the value of f at the point t = 1). Show that E₁ is a linear transformation and find the matrix AE, repre- senting it w.r.t. the standard bases. (6) (Mƒ)(t) = (tf)'. Show that M is a linear transformation and find the matrix AM representing it w.r.t. the standard bases. (7) Are the 4 columns of the representing matrix AM linearly inde- pendent?