M ATH 2S03: L INEAR A LGEBRA III, W INTER 2009
H OMEWORK 7
S OLUTIONS
R EQUIRED PROBLEMS
(1) Let V = Cn be the standard complex n-dimensional vector space equipped with the standard hermitian inner product. Let W V be a k-dimensional subspace of V , with
M ATH 2S03: L INEAR A LGEBRA III, W INTER 2009
T UTORIAL WORKSHEET
PARTIAL SOLUTIONS
(1) You know from 2R03 the denition of the trace tr(A) of any n n matrix A: it is the sum of the
diagonal entries. Now dene the real vector space su(2) as
su(2) := cfw_A
M ATH 2S03: L INEAR A LGEBRA III, W INTER 2009
I N - CLASS T EST # 2
S OLUTIONS
1. (25 marks) Recall that youve shown already that the Pauli spin matrices
i1 = i
0
1
1
0
,
i
0
0
i
i2 = i
,
i3 = i
1
0
0
1
,
form a basis for su(2) as a real vector space. Re
M ATH 2S03: L INEAR A LGEBRA III, W INTER 2009
H OMEWORK 8
S OLUTIONS
Required problems:
All notation and terminology follows the lectures.
1. Recall that Kirkhoffs Current Law states that, for a solution I C1 to the network problem, the
algebraic sum of
M ATH 2S03: L INEAR A LGEBRA III, W INTER 2009
T UTORIAL P ROBLEMS : W EEK 3
PARTIAL S OLUTIONS
For this sequence of problems, you may assume that the pairings given in class on L2 ([L, L], C) (or
L ([L, L], R) are indeed a hermitian (respectively real) i
M ATH 2S03: L INEAR A LGEBRA III, W INTER 2009
T UTORIAL P ROBLEMS
PARTIAL S OLUTIONS
Here are some practice problems with linear algebra over F =
mension etc.) are over the scalars F = Z/2Z.
Z/2Z. All notions (such as basis, di-
1. Compute the number of
McMASTER UNIVERsiTY
MATH 233 TEST/f
Dr. M. Wang
DATE: March 13, 2013.
DURATION OF TEST: 85 minutes
FAMILY NAME: _ FIRST NAME:
STUDENT NUMBER:
_ 10
This test paper should contain 4 questions with parts on a total oil/g pages. You
are responsible for ensu
M ATH 2S03: L INEAR A LGEBRA III, W INTER 2009
T UTORIAL P ROBLEMS
PARTIAL SOLUTIONS
This sequence of problems explores in some more detail the spectral theory on hermitian vector spaces
which we have been discussing in class. In particular, it will ask y