Math 312 Solutions #7
1. Dene an inner product on R[x]3 by
1
f, g =
1
f (x)g(x)
dx .
1 x2
For 0 j 3, let Dj End R[t]3 be the operator dened by
Dj (f ) = (1 x2 )f (x) xf (x) + j 2 f (x) .
(a) Apply the Gram-Schmidt procedure to the basis (1, x, x2 , x3 ) t
Math 312 Solutions #9
1. Suppose that T End(V ) is self-adjoint, K and > 0. Prove that if there
exists v V such that v = 1 and T v v < , then T has an eigenvalue
such that | | < .
Hint: Express v in terms of the orthonormal eigenbasis given by the spectr
Problem Set #6
Due: Friday, February 16, 2007
1. Suppose T End(V ) has dim V distinct eigenvalues and that S End(V ) has
the same eigenvectors as T (not necessarily with the same eigenvalues). Prove
that ST = T S.
2. Let V be a complex inner product space
Problem Set #7
Due: Friday, March 2, 2007
1. Let n = 3. Dene an inner product on R[x]n by
1
f, g :=
1
f (x)g(x)
dx .
1 x2
(a) Apply the Gram-Schmidt procedure to the basis (1, x, . . . , xn ) to produce an
orthonormal basis e0 (x), . . . , en (x) of R[t]n
Problem Set #8
Due: Friday, March 9, 2007
1. Let f (x) = 1 x + |x| = x if x < 0 . Find the projection of f onto U := R[x]2 using
2
0 if x 0
the following:
1
(a) Legendre polynomials and f, g = 1 f (x)g(x) dx;
1
(x)g(x)
(b) Chebyshev polynomials and f, g =
Problem Set #9
Due: Friday, March 16, 2007
1. Suppose S End(V ) satises S 2 = S. Prove that S is an orthogonal projection
if and only if S is self-adjoint.
2. Suppose that T End(V ) is self-adjoint, K and > 0. Prove that if there
exists v V such that v =
Problem Set #5
Due: Friday, February 9, 2007
1. Suppose that a0 , . . . , am are distinct elements in K and that b0 , . . . , bm are elements in K. Prove that there exists a unique polynomial p K[t]m such that
p(aj ) = bj for 0 j m.
2. Consider T Hom(R[x]
Problem Set #4
Due: Friday, February 2, 2007
1. Let L : C 2 ([0, 1]) C([0, 1]) be dened by Lf = f .
(a) Show that L has no left inverses.
Hint: L is not injective.
(b) Show that the operators G1 and G2 , dened as follows, are right inverses:
x
(x t)f (t)
Problem Set #3
Due: Friday, January 26, 2007
1. The conjugate transpose of a complex (m n)-matrix Z is the (n m)-matrix Z
obtained by interchanging the rows and columns and then taking the complex
conjugate of each entry. A complex (n n)-matrix Z is Herm
Problem Set #2
Due: Friday, January 19, 2007
1. The transpose AT of an (m n)-matrix A is obtained from A by interchanging
the rows with the columns; in other words if A = [ai,j ] then AT = [aj,i ]. A matrix
A is symmetric if AT = A and skew-symmetric if A
Problem Set #10
Due: Friday, March 23, 2007
1. (a) Establish that any nonnegative linear combination of positive operators is
positive.
(b) Suppose T End(V ) is positive. Prove that T k is positive for all positive
integers k.
2. Let V be the R-vector spa
Problem Set #11
Due: Friday, March 30, 2007
1. Suppose that T End(V ) has a singular-value decomposition given by
T v = s1 v, u1 e1 + + sn v, un en
for all v V , where s1 , . . . , sn are the singular values of T and (u1 , . . . , un ),
(e1 , . . . , en )
Problem Set #12
Due: Wednesday, April 4, 2007
1. (a) Find the Jordan Canonical Form of the matrix A =
1
0
0
0
0
1
0
1
1
1
1
0
0
0
1
0
1
0
0
0
0
1
1
0
0
0
0
0
1
0
0
0
0
0
0
1
.
(b) Let T End(Cn ). Suppose that the characteristic polynomial f of T and
2
min
Math 312 Solutions #10
1. (a) Show that any nonnegative linear combination of positive operators is
positive.
Solution. Let V be an inner-product space. If S1 , . . . , Sr are positive operators and a1 , . . . , ar are nonnegative real numbers, then we ha
Math 312 Solutions #6
1. Suppose T End(V ) has dim V distinct eigenvalues and that S End(V ) has
the same eigenvectors as T (not necessarily with the same eigenvalues). Prove
that ST = T S.
Solution. Let dim V = n and let (v1 , . . . , vn ) denote the eig
Math 312 Solutions #11
1. (a) Give a counterexample to the following statement: if T End(V ) then
V = Ker T Im T .
Solution. Let T : R2 R2 be given by T (x, y) = (y, 0). The image of T
is the same as the kernel of T : Im T = Ker T = cfw_(x, 0) | x R. Thus
Math 312 Solutions #3
1. The conjugate transpose of a complex (mn)-matrix Z is the (nm)-matrix Z
obtained by interchanging the rows and columns and then taking the complex
conjugate of each entry. A complex (n n)-matrix Z is Hermitian (or selfadjoint) if
Math 312 Solutions #4
1. Let T : C 2 ([0, 1]) C([0, 1]) be dened by T f = f .
(a) Show that T has no left inverses.
(Hint: Show that T is not injective.)
(b) Show that S, dened as follows, is a right inverse of T :
x
(x t)f (t) dt
(Sf )(x) =
0
(Hint: Use
Math 312 Solutions #5
1. Suppose that a0 , . . . , am are distinct elements of k and that b0 , . . . , bm are any
(not nescessarily distinct) elements of k. Prove that there exists a unique
polynomial p k[t]m such that p(aj ) = bj for 0 j m.
Hint: Examine
Math 312 Solutions #1
1. Give an example of a nonempty subset U of R2 such that U is closed under
scalar multiplication, but U is not a subspace of R2 .
Solution. Let U := cfw_(x, y) R2 : xy = 0; in other words, U is the union of the
x-axis and the y-axis
Math 312 Solutions #2
1. The transpose AT of an (m n)-matrix A is obtained from A by interchanging
the rows with the columns; in other words if A = [ai,j ] then AT = [aj,i ]. A
matrix A is symmetric if AT = A and skew-symmetric if AT = A.
(a) Prove that t
Problem Set #1
Due: Friday, January 12, 2007
1. Give an example of a nonempty subset U of R2 such that U is closed under
scalar multiplication, but U is not a subspace of R2 .
2. Let K be any eld and let KK denote the set of all functions from K to K. The