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MATH 245 Midterm
Prof. K.R. Davidson
Tuesday, November 6, 2012 4:306:30
Marks
[8]
1. (a) Define the characteristic polynomial pT of T L(V ).
(b) State the Cayley Hamilton Theorem.
(c) What is the kernel of the map from F[x] to A(
Math 245, Spring 2012
Assignment 5
Due Friday, June 15, in class.
1. Let J be a Jordan block with eigenvalue . Let A = etJ , where t C.
(a) Prove that
(A0 )ij =
tj i /(j i)!
0
if i j
otherwise .
(b) In general, prove that
(A )ij =
et tj i /(j i)!
0
if i j
Math 245, Spring 2012
Assignment 10
Due Friday, July 20, in class.
1. Let x, y V , where V is a nite dimensional vector space over a eld F. Prove that x y =
y x in V V if and only if x and y are linearly dependent.
2. For each a C let Ma : C C be the mult
Math 245, Spring 2012
Assignment 11
Not to be handed in.
1. Let cfw_v1 , v2 , . . . , vn be a basis for a vector space V over a eld F. Let cfw_f1 , f2 , . . . , fn be the
dual basis.
(a) Suppose 1 i1 < i2 < < ik n and 1 j1 < j2 < < jk n. Show that
fj1 f
Math 245, Spring 2012
Assignment 9
Not to be handed in.
1. Let A Mnn (R) be a skew-symmetric matrix. Show that eA is an orthogonal matrix. (You
may use the fact that eX +Y = eX eY if XY = Y X .)
2. Let Q Mnn (R) be an orthogonal matrix.
(a) Prove that det
Math 245, Spring 2012
Assignment 8
Due Friday July 6, in class.
1. Let A Mnn (F) be a self-adjoint matrix (where F = R or F = C). Prove that A is
positive-denite if and only if all eigenvalues of A are positive real numbers.
2. Let V be an n-dimensional i
Math 245, Spring 2012
Assignment 7
Due Friday, June 29, in class.
Throughout this assignment, the eld F is either R or C.
1. A self-adjoint matrix A Mnn (F) is said to be positive-denite if the map : Fn Fn F,
(x, y ) = y Ax is positive-denite.
Illustrate
Math 245, Spring 2012
Assignment 6
Due Friday, June 22, in class.
Throughout this assignment, the eld F is either R or C.
1. Let V be a vector space over F. Let , , and , be two inner products on V . For R,
dene
x, y = x, y + x, y .
(a) If > 0, prove that
Math 245, Spring 2012
Assignment 3
Due Friday, June 1, in class.
1. A linear operator T is said to be nilpotent if there exists a positive integer m such that T m
is the zero operator.
Let V be a vector space over the eld of complex numbers, and let T L(V
Math 245, Spring 2012
Assignment 2
Due Friday, May 25, in class.
In problems 13, the quotient map V V /W is denoted by v v + W .
1. Let V = R[x] (as a vector space over R), and let T L(V ) be the linear operator T (f (x) =
(x2 + 1)f (x). Let W = im(T ).
(
Math 245, Spring 2012
Assignment 1
Due Friday, May 18, in class.
1. District of Columbia is a two player game. At each point in the game, one player is the
President. The President gets to sit in a padded armchair and carry a large stick. The
other player
MATH 245 Linear Algebra 2, Solutions to Assignment 1
2
1
3
1
1
1
1
2
4
2
1
2
1: (a) Let p =
, u1 =
, u2 = , q =
, v1 = and v2 =
.
2
1
1
3
4
1
1
3
5
0
2
1
Find the point of intersection of the plane x = p + t1 u1 + t2 u2 and the plane x = q
MATH 245 Linear Algebra 2, Assignment 1
Due Fri May 23
2
1
3
1
1
1
1
2
4
2
1
2
1: (a) Let p = , u1 = , u2 = , q = , v1 = and v2 = .
2
1
1
3
4
1
1
3
5
0
2
1
Find the point of intersection of the plane x = p + t1 u1 + t2 u2 and the plane x = q +
MATH 245 Linear Algebra 2, Assignment 2
3
1: Let A = 2
1
Due Fri June 6
1 2
4 4 M3 (R). Diagonalize A, if possible.
1 0
3 1 3
2: Let A = 2 4 6 M3 (R) and dene L : R3 R3 by L(x) = Ax. Let U be the subspace of R3 with
2 1 3
basis U = cfw_u1 , u2 where u1 =
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MATH 245 Linear Algebra 2, Solutions to the Midterm Test, Spring 2013
[5]
1: Let A and B be non-empty convex sets in Rn . Let A + B = a + ba A, b B . Show
that A + B is convex.
Solution: Let u, v A + B, say u = a + b and v = c
University of
Math 245
WATERLOO
Final Examination
Linear Algebra II
Fall 2011
Instructor: Kenneth R. Davidson
Section: 001
Term: 1119
Materials: Ordinary scientic calculators
Date: Saturday, December 17, 2011
Time: 9:00 a.m. to 11:30 a.m.
Duration of exam
MATH 235 Midterm Solutions Fall 2005
1. (a) [2] Given that det(A) = 2 and det(B) = 5, determine dct(A3B1ATB2).
52g: : dLHA'sBwgzx =1di+<mj5d1+mwdi+<mjl
OMB»
(b) [3] Let A be an n x n matrix. A is called skew-symmetric if AT 2 A. Prove
that if A is s
Family Name (Print):
Signature: :
ID. No.: #
Math 245 Midterm Examination
Tuesday Feb. 8, 2005
NO ADDITIONAL MATERIAL ALLOWED
Instructor C.T.Ng
Instructions:
1. Put your name, signature and ID. No. in the blanks above.
2. This exam contains 8 pages
MATH 245 Linear Algebra 2, Lecture Notes, by Stephen New
1. Ane Spaces, Convex Sets and Simplices
1.1 Note: In this section, we let F denote a xed eld (for example F = Q, R, C or Zp
with p prime) and let W be a xed vector space over F .
1.2 Denition: An a
MATH 245 Linear Algebra 2, Solutions to Assignment 2
3
1: Let A = 2
1
1 2
4 4 M3 (R). Diagonalize A, if possible.
1 0
Solution: The characteristic polynomial of A is
3x
fA (x) = det(A xI) = det 2
1
1 2
4 x 4
1 x
= x(x 3)(x 4) 4 4 4(x 3) + 2x 2(x 4)
= (x3
MATH 245 Linear Algebra 2, Assignment 3
3 1 1
1 0 0
1: Let A =
0 1 1
1 3 1
Due Fri June 27
0
1
M4 (R). Put A into Jordan Form, if possible.
1
4
2: (a) Let A M10 (Q). Suppose the characteristic polynomial is fA (x) = (x 2)7 (x 3)3 and the minimal
polynom
MATH 245 Linear Algebra 2, Solutions to the Midterm Test
[5]
1 4 2
1: Let A = 1 3 1 .
1 2 0
Find an invertible matrix P and a diagonal matrix D such that P 1A P = D.
Solution: The characteristic polynomial of A is
1x
4
2
fA (x) = det 1
3x 1
1
2
x
= x(x 1
Math 245, Spring 2012
Assignment 4
Not to be handed in.
1. Let V be a nite dimensional vector space, T L(V ).
(a) Let W be a T -invariant subspace of V . Prove that the minimal polynomial of TW divides
the minimal polynomial of T .
(b) Suppose V = W1 Wk ,
MATH 245 Linear Algebra 2, Assignment 5
Due Wed Nov 20
2 2 2
T
1: (a) Let A = 2 1 4 . Find an orthogonal matrix P and a diagonal matrix D such that P AP = D.
2 4 1
(b) Let A =
1+i
2i
1i
. Find a unitary matrix P and an upper-triangular matrix T so that P
MATH 245 Linear Algebra 2, Assignment 4
Not to hand in
1
1
2
1
1
3
1
0
1: Let u1 = , u2 = , u3 = and x = . Let U = cfw_u1 , u2 , u3 and let U = Span U . Find
7
2
1
1
3
1
0
1
ProjU (x) in the following three ways.
(a) Let A = u1 , u2 , u3 M43 then use th
MATH 245 Linear Algebra 2, Assignment 3
Due Fri June 21
12
34
3 1
, A2 =
and A3 =
. Apply the Gram-Schmidt Procedure to the
1 0
1 2
2 4
basis U = A1 , A2 , A3 to obtain an orthonormal basis for U = Span A1 , A2 , A3 M22 (R).
1: (a) Let A1 =
(b) Find an or
MATH 245 Linear Algebra 2, Solutions to Assignment 1
2
1
3
1
1
1
1
2
4
2
1
2
1: (a) Let p =
, u1 =
, u2 = , q =
, v1 = and v2 =
.
2
1
1
3
4
1
1
3
5
0
2
1
Find the point of intersection of the plane x = p + t1 u1 + t2 u2 and the plane x = q + s1 v1 +
MATH 245 Linear Algebra 2, Solutions to Assignment 1
1: Let P = a + Null(A) and Q = b + Col(B ) where
3
1
1 1
1
2
a = , b = , A = 3 1
2
1
10
4
3
1
0 1
1
2 1 , B=
2
11
0
12
0 3
.
42
3 1
Find a point p R4 and a basis for a subspace U R4 such that P Q = p +