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 giv
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 th
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
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 eig
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,
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 ov
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
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 t
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 (
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 basi
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 par
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
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 bas
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
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 t
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)
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 b
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 nxr
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 polynomi
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
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
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,
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) =
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).
Sol
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