Math 108B - Take-Home Midterm Solutions
1. The matrix 2 11 42
represents a linear transformation T : R2 R2 with respect to the basis cfw_v1 , v2 where v1 = (3, 1) and v2 = (0, 2). Find the matrix of T with respect to the basis cfw_w1 , w2 where w1 = (1,
Math 108B - Home Work # 4 Solutions LADR Problems p. 125 24. Notice that (p) = p(1/2) is a linear functional on P2 (R). Thus we follow the idea 1 of the proof of 6.45 to nd a polynomial q P2 (R) such that (p) = p, q = 0 p(x)q(x) dx for all p. need an orth
Math 108B - Home Work # 6
Due: Wednesday, June 4, 2008
1. If A is an n n upper-triangular matrix (i.e., Aij = 0 for all i > j), show that det A = n Aii . (You may use either the denition of determinant given in class, or i=1 else the standard denition for
Math 108B - Home Work # 3
Due: Wednesday, April 23, 2008
1. LADR p. 123-124: Exercises 10, 14, 15, 21 (For 14, rst read Corollary 6.27).
2. (Do not turn in) For Ex. 10, does anything change if you apply the Gram-Schmidt
1
Process to the basis cfw_1, x, x2
Math 108B - Home Work # 2
Due: Wednesday, April 16, 2008
1. Let b1 , . . . , bn be positive real numbers. Check that the form
z, w = b1 z1 w1 + bn zn wn
denes an inner product on F n , where z = (z1 , . . . , zn ) and w = (w1 , . . . , wn ). (In
particula
Math 108B - Home Work # 5
Due: Friday, May 16, 2008
1. LADR p. 159-160: Exercises 11, 13 (Hint: diagonalize), 15, 22.
2. Recall that if U is a subspace of the inner product space V , we dened the reection
in U to be the linear map RU : V V given by
RU = 2
MATH 10813 Midterm # 2 F 2012
Nov. 19, 2012 Instructor: X. Dad
Name _
All questions have equal points. Show complete work. Continue on the back of
the page if you need more space. Good luck?
1. Let V = R2 be given the inner product (3:, y) = 4531311 +
Solutions
Homework 5
7.1. Make P2 (R) into an inner-product space by dening
1
p, q =
p(x)q(x)dx.
0
Dene T L(P2 (R) by T (a0 + a1 x + a2 x2 ) = a1 x .
(a) Show that T is not self-adjoint.
(b) The matrix of T with respect to the basis (1, x, x2 ) is
0 0 0
Math 108B - Take-Home Midterm 2 Solutions
1. Let V be a nite-dimensional vector space. We dened the dual space of V as the
vector space V = L(V, F ) of linear functionals on V . We write V for the dual space
of V .
(a) For v V , dene v : V F by v (f ) = f
Math 108B - Home Work # 1
Due: Friday, April 11, 2008
1. Let T : R2 R3 be the linear transformation given by the matrix
1 1
2
2
0
3
with respect to the standard bases. Find bases for R2 and R3 in which the matrix of
T is
1 0
0 1
0 0
2. The matrix
4 1
Math 108B - Home Work # 5 Solutions 1. LADR Problems, p. 159-160: 11. Let T be a normal operator on the complex inner-product space V . By the spectral theorem there is an orthonormal basis cfw_e1 , . . . , en of V consisting of eigenvectors T . for If T
Math 108B - Home Work # 6 Solutions 1. If A is an n n upper-triangular matrix (i.e., Aij = 0 for all i > j), show that det A = n Aii . i=1 Solution. As done in class, we can compute the determinant of A by simplifying the wedge product of the columns of A
Math 108B - Home Work # 3 Solutions 1. LADR Problems. 10. We have |1| =
1 0
12 dx = 1, so we can take e1 = 1. Now let
1
u2 = x x, e1 e1 = x x, 1 1 = x
0
xdx = x 1/2. 12x 12/2. Now let
Since |u2 | =
1 (x 0
1/2)2 dx =
1/12, we let e2 =
u3 = x2 x2 , e1 e1
Math 108B - Home Work # 1 Solutions 1. For T to have the matrix 10 0 1 00
with respect to a basis cfw_u1 , u2 of R2 and a basis cfw_v1 , v2 , v3 for R3 , means simply that T u1 = v1 and T u2 = v2 . Hence cfw_u1 , u2 can remain the standard basis, and t
Math 108B - Take-Home Midterm
Due: May 23, 2008
Instructions and Rules:
You may use your notes and the texts on this exam. In addition, you may cite any result from lecture, homework problems, or the sections of the text we have covered without proof. Ju
Math 108B - Take-Home Final
Due: 12 pm on June 11, 2008
Instructions and Rules:
You may use your notes and the texts on this exam. In addition, you may cite any result from lecture, homework problems, or the sections of the text we have covered without p
Math 108B - Home Work # 2 Solutions
1. Let b1 , . . . , bn be positive real numbers. Check that the form
z , w = b1 z1 w1 + bn zn wn
denes an inner product on F n , where z = (z1 , . . . , zn ) and w = (w1 , . . . , wn ). (In
particular, the dot product o