MATH 3912 - Assignment 5
1. Suppose p∈P2and p(x0) = p0(x0) = p00(x0) = 0. Show that pmust be identically zero.
2. Suppose p∈P2and suppose that p(x1) = p0(x1) = 0 and p(x2) = 0 for some fixed numbers x16=x2.
Show that pmust then be identically zero.
3. Show that if p∈Pnand phas a root of total multiplicity n+ 1 then pmust be identically zero. Recall
that phas a root of multiplicity n+ 1 at x=aif p(a) = p0(a) = . . . =p(n)(a) = 0
4. If p(x) is a polynomial of even degree, what is limx→∞p(x) and limx→−∞p(x)? Evaluate the same limits
in case pis a polynomial of odd degree.
5. If f(x) is a polynomial then limn→∞f(n)(x) = 0 for all x.
6. Show that f(x) = 2xcan coincide with a polynomial at only a finite number of points.
7. Suppose fis real analytic for all xand f(k)(x)>0 for k= 0,1, . . .. Then fcan not coincide with a
polynomial infinitly often.
8. Suppose X=C1([a, b]) and x0is a fixed point in [a, b]. Define L(f) = f0(x0) for all f∈X. Is La linear
functional?
9. Suppose X=C0([a, b]). Define two functionals Land Gvia
L(f) = Zb
a
x2f(x)dx and G(f) = Zb
a
(f(x))2dx
Which of these functions is linear?
10. Suppose X=Pn,x0is a fixed point in [a, b], and fis a continous function defined on the interval [a,b].
Define L(p) = (f◦p)(x0). Is La linear functional?
11. Suppose X=C1([a, b]). Define L(f) = f0for all f∈X. What is the range of this map L? Is the map
Llinear? Is La linear functional?
12. Let Xbe the space of n×nmatrices and define L(A) = det(A). Is La linear functional?
13. Let Xbe the space of n×nmatrices. For a matrix A∈Xdefine the trace of A as
trace(A) = trace
a1,1a1,2. . . a1,n
a2,1a2,2. . . a2,n
.
.
..
.
.....
.
.
an,1an,2. . . an,n
=a1,1+a2,2+. . . +an,n
Is L(A) = trace(A) a linear operator?