**Q1: **

Consider the vector space of polynomials of degree three at most. The differentiation operator is a linear transformation on this vector space. Find the matrix that represents the linear transformation with respect to the basis .

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**Q2: **

Consider the vector space of polynomials of degree three at most. The differentiation operator is a linear transformation on this vector space. Find the matrix that represents the linear transformation with respect to the basis .

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**Q3: **

Consider the vector space of infinitely differentiable functions. The differentiation operator is a linear transformation on this vector space. Find the general form of an element of the kernel of the linear transformation .

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**Q4: **

Consider the vector space of infinitely differentiable functions. The differentiation operator is a linear transformation on this vector space. Find the general form of an element of the kernel of the linear transformation .

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**Q5: **

Apply the linear differential operator to evaluate the following expression: .

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