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 for IR2 . The matrix Q diagonalizes A, where Q= 1 1 2 3
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 are solutions too. (e) True: directly from linearity and
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 = [dij ] are connected by the formulas
n n
cij =
k=1
aik b
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 map T is 1 1 as we see that T (a(1, 0) + b(1, 1) = a(1
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) Yes, the corresponding linear system has a solution a
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 = c x, y .
¯
(b) If x, y, and z are vectors in V such tha
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 and b. (d) False: take a to be the zero scalar and any
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 matrices always have the same eigenvalues.
(Solution) True.
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.
(Solution) True : M has rank n if and only if M is inver
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 true or false. No
veriﬁcation is needed.
(a) Given x1
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 determinant. (e) True (proved in this section). Prob 4.1.2. (a
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 IN. Since the determinant of the zero matrix is zero,
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 IN. Since the determinant of the zero matrix is zero,
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 determinant. (e) True (proved in this section). Prob 4.1.2. (a
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 are solutions too. (e) True: directly from linearity and
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 = [dij ] are connected by the formulas
n n
cij =
k=1
aik b
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 map T is 1 1 as we see that T (a(1, 0) + b(1, 1) = a(1
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) Yes, the corresponding linear system has a solution a
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 and b. (d) False: take a to be the zero scalar and any
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 conjugate linear in the second component. (d) False, there a
Solutions to Homework 11.
Math 110, Fall 2006. Prob 5.4.1. (a) False, cfw_0 and the whole space are T -invariant for any T . (b) True (Theorem 5.21). (c) False: just take w to be a nonzero multiple of v, then the corresponding T -cyclic subspaces are the
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 for IR2 . The matrix Q diagonalizes A, where Q= 1 1 2 3
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 space V , then span(S) equals the
intersection of all sub