Well, L(p)= 0, not L(p'). You must solve xp'(x)= 0 for all x.Niles said:Homework Statement
If I e.g. want to find the kernel and range of the linear opertor on P_3:
L(p(x)) = x*p'(x),
then we can write this as L(p'(x)) = x*(2ax+b). What, and why, is the kernel and range of this operator?
The Attempt at a Solution
The kernel must be the x's where L(p'(x)) = 0, so do I just solve the equation?
If every member of P_3 can be written as [itex]p(x)= ax^2+ bx+ c[/itex], then [itex]p'(x)= 2ax+ b[/itex] and so L(p(x))= [itex]2ax^2+ bx[/itex]. That set of functions is the range. Since a can be any number, so 2a can be any number.The range is the elements that span the polynomial, so it is span{(x^2,x)}?
The kernel of a linear operator is the set of all inputs that are mapped to the zero vector by the operator. In other words, it is the set of all vectors that when multiplied by the operator result in the zero vector.
The image of a linear operator is the set of all possible outputs that can be obtained by applying the operator to the input vectors. It is also known as the range of the operator.
Yes, one example of a linear operator is the transformation matrix used in a rotation or translation in 3D space. Another example is the differentiation operator in calculus.
The kernel and image of a linear operator can be used to solve systems of linear equations, find eigenvalues and eigenvectors, and determine whether a linear transformation is invertible. They are also important in applications involving vector spaces, such as computer graphics and control systems.
No, the size of the kernel and image of a linear operator can vary depending on the dimensions of the input and output spaces. In general, the kernel and image can have different sizes, but the dimension of the image cannot be larger than the dimension of the input space.