The following at linear transformations. Find the kernel and images:

1) φ: P4 ->R given byφ(p(x)) =p′′(0), wherep′′(x) denotes the second derivative

2)φ:V ! Vgiven byφ(f) =f′+f, whereVis the subspace of the space of smooth functions R!R spanned by sin and cos, andf′denotes the derivative