Solutions to Homework 10.
Math 110, Fall 2006. Prob 5.1.3. (a) The eigenvalues are 1 and 4, with the eigenspaces E1 = spancfw_[1 1]t , E4 = t spancfw_[2 3] . The vectors [1 1]t , [2 3]t form a basis f
Solutions to Homework 2.
Math 110, Fall 2006. Prob 1.4.4. (a) Yes, since the linear system a+b 2a + 3b -a a-b has a solution a = 3, b = -2. (b) No, the corresponding linear system has no solution. (c)
Quiz #5
1. Label the following statements as being true or false. Assume that V
is a ﬁnite-dimensional inner product space
(a) An inner product is linear in both components.
(Solution) False. x, cy =
Solutions to Homework 1.
Math 110, Fall 2006. Prob 1.2.1. (a) True this is axiom (VS 3). (b) False we proved the zero vector is unique. (c) False: take x to be the zero vector, and take any scalars a
Quiz #4
1. Label the following statements as being true or false.
(a) Any two eigenvectors are linearly independent.
(Solution) False. Consider an constant multiple of an eigenvector.
(b) Similar matr
Quiz #3
1. Label the following statements as being true or false. All matrices in the
problems are square. No veriﬁcation is needed.
(a) A matrix M ∈ Mn×n (F ) has rank n if and only if det(M ) = 0.
(
Quiz #2
1. Let V and W be ﬁnite-dimensional vector spaces (over F ) with ordered
bases α and β, respectively. And let T be a linear transformation from
V to W . Label the following statements as being
Quiz #1
1. Label the following statements as being true or false. No veriﬁcation
is needed.
(a) A vector space may have more than one zero vector.
(Solution) False.
(b) If S is a subset of a vector sp
Homework Set # 15
PHYS 110: Electrodynamics Core
Fall 2015
Due Friday, Dec. 11, in class.
Homework Problems: (unless otherwise stated, all problems worth 5 points)
1. (15 pts.) As asked by one of your
Phys 110 HW#1 solns
1
1.) Given a triangle (NOT necessarily a "right triangle") with sides a, b, and c, and an angle
() opposite side c. Suppose I tell you a, b, and , and ask "what is c?"
-> do you k
Electrostatics Review
Physics 110
December 8, 2015
This is an updated version of the midterm review material. Blue text indicates background material,
especially from the first half, which is consider
Solutions to Homework 3.
Math 110, Fall 2006. Prob 2.1.10. By the linearity of T , we use the fact (1, 0) + 3(1, 1) = (2, 3) to obtain T (2, 3) = T (1, 0) + 3T (1, 1) = (1, 4) + 3(2, 5) = (5, 11). The
Solutions to Homework 4.
Math 110, Fall 2006. Prob 2.3.13. Let A = [aij ]. Then At = [aji ], and
n n
tr(A) =
i=1
aii =
j=1
ajj = tr(At ).
The elements of A = [aij ], B = [bij ], AB = [cij ] and BA = [
Solutions to Homework 9.
Math 110, Fall 2006. Prob 4.3.10. Since det(AB) = det(A) det(B) for any two square matrices of the same order, this implies, by induction, that (det M )k = det(M k ) for all k
Solutions to Homework 8.
Math 110, Fall 2006. Prob 4.1.1. (a) False, it is 2-linear. (b) True. (c) False, A is invertible if and only if det(A) = 0. (d) False, it is the absolute value of that determi
Solutions to Homework 7.
Math 110, Fall 2006. Prob 3.1.3. To find the elementary operations: 0 (a) 0 1 inverses of the elementary matrices, we must simply undo the corresponding 0 1 0 1 0 , 0 (b) 1 0
Solutions to Homework 6.
Math 110, Fall 2006. Prob 2.7.1. (a) True. (b) True. (c) False: the roots of the auxiliary polynomial give frequencies of solutions. (d) False: linear combinations of these ar
Solutions to Homework 4.
Math 110, Fall 2006. Prob 2.3.13. Let A = [aij ]. Then At = [aji ], and
n n
tr(A) =
i=1
aii =
j=1
ajj = tr(At ).
The elements of A = [aij ], B = [bij ], AB = [cij ] and BA = [
Solutions to Homework 3.
Math 110, Fall 2006. Prob 2.1.10. By the linearity of T , we use the fact (1, 0) + 3(1, 1) = (2, 3) to obtain T (2, 3) = T (1, 0) + 3T (1, 1) = (1, 4) + 3(2, 5) = (5, 11). The
Solutions to Homework 2.
Math 110, Fall 2006. Prob 1.4.4. (a) Yes, since the linear system a+b 2a + 3b -a a-b has a solution a = 3, b = -2. (b) No, the corresponding linear system has no solution. (c)
Solutions to Homework 1.
Math 110, Fall 2006. Prob 1.2.1. (a) True this is axiom (VS 3). (b) False we proved the zero vector is unique. (c) False: take x to be the zero vector, and take any scalars a
Solutions to Homework 12.
Math 110, Fall 2006. Prob 6.1.1. (a) True, directly by definition. (b) True, this is also required by the definition. (c) False, it is linear in the first component and conju