Let D be the differentiation operator on P3, and let S = {p ∈ P3 | p(0) = 0} Show that (a) D maps P3 onto the subspace P2, but D: P3 → P2 is not one-to-one. (b) D: S → P3 is one-to-one but not onto.

Give a counter example to show that W is not subspace of R³. W=set of (x, y, z) where x + y = 1

Consider the vector space F of all functions from R to R with pointwise addition and scalar multiplication. (a) Show that the three vectors fi = e“, f2 = xe", f3 = x²e" are linearly independent in F. (b) Consider the subspace H = span{f1, f2, f3}. Let T : H → H be the linear transformation defined by df T(f)= + f. dx Find the matrix of T' with respect to the basis B = {fi, f2, f3} of H (i.e., find 8[T]B). (c) Is there a basis C of H in which c |T|c is diagonal? Explain why or why not.

use the attached image to graph the linear subspace defined by the kernel.

Verify that the set C[a, b] of all continuous real-valued functions defined on the interval a ≤ x ≤ b is a vector space, with addition and numerical multiplication defined by (f+g)(x) = f(x) + g(x) and (tf)(x) = tf(x).