Math 562 - Homework 4
Solutions
Prof. Arturo Magidin
1. Let V = R3 with its usual inner product, and let L : V V be given by
L(a, b, c) = (3a + b, b c, 5a + c).
Find an expression for L (a, b, c).
Answer. The coordinate matrix of L relative to
3
[ L] = 0
Math 562 - Homework 3
Solutions
Prof. Arturo Magidin
1. Let V = C3 with the usual inner product. Verify that [b1 , b2 , b3 ] are an orthogonal (not
necessarily orthonormal) basis, where b1 = (1, 1, 1), b2 = (2, 1, 1), and b3 = (0, i, i). Then use
inner pr
Math 562 - Homework 2
Solutions
Prof. Arturo Magidin
1.
(i) Apply the Gram-Schmidt orthonormalization process to S in the inner product space R3
(equipped with the standard inner product) to obtain an orthonormal basis :
S = (1, 0, 1), (0, 1, 1), (1, 3, 3
Math 562 - Homework 1
Solutions
Prof. Arturo Magidin
1. Let V be an inner product space over F . Prove the polar identities :
(a) If F = R, then x, y = 1 x + y 2 1 x y 2 .
4
4
Proof. Use the fact that v 2 = v, v and expand the right hand side:
x+y
2
xy
2
MATH 562
FINAL EXAM SOLUTIONS
Prof. Arturo Magidin
1. Find the singular values of the matrix A, where
1
0
A=
1
1
(10 points)
1
1
.
0
1
Solution. The singular values of A are the square roots of the nonzero eigenvalues of A A. So
rst we compute A A:
11
32
MATH 562 Linear Algebra and Applications II
Minimizing dissonance in a plucked string
We saw in class that the solution to the wave equation
2f
1 2f
2 2 = 0,
x2 v t
on [0, L], with Dirichlet boundary conditions f (0, t) = f (L, t) = 0 for all t; and init
MATH 562 MIDTERM
Solutions
Prof. Arturo Magidin
1. Let V be a nite dimensional complex inner product space, and let T be a linear operator on V.
For each of the following types of operators, give a property that characterizes that type in terms
of the exi
Math 562 - Homework 9
Solutions
Prof. Arturo Magidin
1. We found in the last homework the following singular value decomposition:
2
i 22
1+i
1
6 0 3(1i)
2
3
=
A=
1 i i
00
2
3
i 2
2
2
3
3
3
3(1+i)
3
Use this decomposition to nd the pseudoinverse A of A.
.
Math 562 - Homework 8
Solutions
Prof. Arturo Magidin
2
1. Let V = W = span(1, sin x, cos x) with inner product f , g = 0 f (t)g (t) dt. Let T : V W
be given by T (f ) = f + 2f . Find orthonormal bases = [v1 , . . . , vn ] and = [w1 , . . . , wm ]
for V an
Math 562 - Homework 7
Solutions
Prof. Arturo Magidin
1. Find a solution to the wave equation (with L = v = 1) with initial condition
f (x, 0) = 0
f (x, 0) =
t
and
1
if x 2 ;
1
if 2 < x 1.
x
1x
(This corresponds to a string which is struck, such as a piano
Math 562 - Homework 6
Solutions
Prof. Arturo Magidin
1. Let T be a linear operator on a complex inner product space, not necessarily nite dimensional,
and assume that T has an adjoint T . Prove:
(a) If T is self-adjoint, then T (v), v is real for all v V.
Math 562 - Homework 5
Solutions
Prof. Arturo Magidin
1. Find the plane z = a + bx + cy that has the best t (in the sense of least squares) to the points
(0, 0, 0),
(0, 1, 0),
Answer. Plugging in the values we obtain
b, and c:
(0, 0, 0) :
(0, 1, 0) :
(1, 0