MATH 110, mock midterm test.
Name
Student ID #
All the necessary work to justify an answer and all the necessary steps of a proof must be
shown clearly to obtain full credit. Partial credit may be given but only for signicant
progress towards a solution.
Partial Solutions for Linear Algebra by Friedberg et al.
Chapter 3
John K. Nguyen
December 7, 2011
3.2.14. Let T, U : V W be linear transformations.
(a) Prove that R(T + U ) R(T ) + R(U )
(b) Prove that if W is nite-dimensional, then rank(T + U ) rank(T )
Partial Solutions for Linear Algebra by Friedberg et al.
Chapter 4
John K. Nguyen
December 7, 2011
4.1.11. Let : M22 (F ) F be a function with the following three properties.
(i) is a linear function of each row of the matrix when the other row is held xe
Partial Solutions for Linear Algebra by Friedberg et al.
Chapter 5
John K. Nguyen
December 7, 2011
5.2.11. Let A be an n n matrix that is similar to an upper triangular matrix and has the distinct eigenvalues 1 , 2 , ., k with corresponding multiplicities
Partial Solutions for Linear Algebra by Friedberg et al.
Chapter 1
John K. Nguyen
December 7, 2011
1.1.8. In any vector space V , show that (a + b)(x + y) = ax + ay + bx + by for any x, y V and any a, b F .
Proof. Let x, y V and a, b F . Note that (a + b)
Partial Solutions for Linear Algebra by Friedberg et al.
Chapter 6
John K. Nguyen
December 7, 2011
6.1.18. Let V be a vector space over F , where F = R or F = C, and let W be an inner product space over
F with product , . If T : V W is linear, prove that
Section 5.2: 15 Suppose that you have a system of dierential equations dx/dt = Ax,
where x(t) is a function taking values in Rn , and A Mnn (R). which is diagonalizable
with eigenvalues 1 , . . . , n , not necessarily distinct. We can nd a basis of n eige
Section 5.4: 15: To prove the Cayley-Hamilton theorem for matrices, we can use the isomorphism between n n matrices Mnn (F ) and linear transformations L(Fn ) given by A LA . The key additional fact that we need about this isomorphism, is that it also pre
Section 7.1: 1 TFFTFFTT Section 7.1: 2b The characteristic polynomial is (t - 4)(t - 1), so the eigenvalues are distinct, and the matrix is diagonalizable. The basis of eigenvectors can be chosen as (2, 3)t , 4 0 and (1, -1)t , leading to a Jordan Form of
Section 5.2
23. We have that
q
p
M j = W1 + W2 = W1 W2 .
Ki +
i=1
(1)
j=1
On the other hand, let i and j be bases of Ki and Mj , respectively. Since W1 = p Ki ,
i=1
the (disjoint) union of i is a basis of W1 . Similarly the union of j is a basis of W2
and
Section 6.1: 16: Showing that H is an inner product space is mostly a straightforward exercise except for the positivity condition. As in example 3, the continuity of the functions is an essential part of the proof. To see why, consider a function that is
May 3, 2013
Section 6.1
15. (b) (i) We show that x + y = x + y if and only if one of x, y is a non-negative
multiple of the other.
() After rearrangement, we may assume that x = cy for some c 0. Then
x + y = (c + 1)y = |c + 1|. y = (c + 1) y = c y + y = c
Section 5.4
20. The direction () is easy, so we only consider the other direction ().
Suppose that dim V = n and V is T -generated by a vector v V . Then the set =
cfw_v, T (v), , T n1 (v) forms a basis of V . There exist 0 , , n1 F such that
U (v) = 0 v
May 18, 2013
Section 6.4
17. Notice that T and U are diagonalizable (self-adjoint) and the eigenvalues are non-negative
(T, U 0 and Exercise (17.a).
(c) [Cf. Exercise 13 of 6.6] () Take an orthonormal basis = cfw_w1 , , wn of V such
that T (wi ) = i wi .
FALL 2013 MATH35300
HOMEWORK 13
DUE: FRIDAY, DECEMBER 6TH
INSTRUCTOR: CHING-JUI LAI
Through this homework, you can always assume that a vector
space is a real vector space. That is the eld F is just the eld
of real numbers R. If you feel comfortable to wo
MATH 110, solutions to the mock midterm.
1. Consider the vector space P (IR) and the subsets V consisting of those vectors (polynomials) f for which:
(a) f has degree 3,
(b) 2f (0) = f (1),
(c) f (t)
0 whenever t
(d) f (t) = f (1
0,
t) for all t.
In which
MATH 110, mock nal test.
Name
Student ID #
All the necessary work to justify an answer and all the necessary steps of a proof must be
shown clearly to obtain full credit. Partial credit may be given but only for signicant
progress towards a solution. Show
M T H 5] 3 Linear Algebra
Chapter 4: Determinants
Theorem 4. 7
Theorem 4.7 For any A, B e Mm (F), det(AB) = det(A)det(B)
Proof:
Case 1: If A is an Elementary Matrix
Defiz: Elementary Matrix An nxrz matrix is considered to be an elementary matrix if it
MTH 513 Chapter 5
Linear Algebra Diagonalization
Quan Ding
Theorem 5.23 (Cayley-Hamilton) Let T be a linear operator on a finite-dimensional vector space V, and let f(t) be the characteristic polynomial of T. Then f(T)=T0, the zero transformation. Corolla
Recall the following Theorems, Propositions and denitions:
Denition: Let T be a linear operator on a vector space V. A subspace
Wof Vis called a Tainvariant subspace of Vif T g; W, that is, if
T(V) E Wfor all V E W.
Denition: Let T be a linear operator on
MTH 513
Dan Hadley
13 November 2007
Denition. Let Vibe a vector space over F. An inner product on V is a function-that
assigns to every ordered pair of vectors 'x and y in V, a scalar in F, denoted <x, y>, such
that for all x, y, and z in V and all c in F
Presentation for Linear Algebra
By Chris Lynd
Denition: Let Vbe an inner product space. For x e V, we dene the norm or
length of x by = W .
Theorem 6.2 Let Vbe an inner product space over F.
Then Vx, y E V and VC 6 F, the following statements are true:
Linear Algebra 1 1-27-2007
PRESENTATION
Theorem 6.9. Let V be a nitedimensional inner product Space, and let T be linear
operator on V. Then there exists a unique mction T I V> V such that
<T(x), y) = <x,T*(y)> for all x, y E V. Furthermore, T" is linear.
Solutions to selected homework problems.
Problem 2.1: Let p be any prime and V = Z2 , the standard twop
dimensional vector space over Zp . How many ordered bases does V have?
Answer: (p2 1)(p2 p).
Solution: First, by Corollary 3.5(c) any basis of V has tw
Solutions to selected problems in homeworks 9-11 (to be continued).
Problem 9.7: Find the Jordan canonical
matrix
210
0 2 1
A=
0 0 3
010
form and a Jordan basis for the
0
0
.
0
3
Solution: Step 1: we compute eigenvalues and multiplicities. Using the
formu