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MATH 196
Homework 21
(Due date: Thursday, April 16)
Please do the following problems. All the cited problem are from Bretscher’s
Linear Algebra with Applications, 4th Edition.
Make sure to show all details of your work.
1. (32pts) Do 2.1.4, 2.1.5, 2.1.6, 2.1.7, 2.1.8, 2.1.17, 2.1.20, 2.1.23 (4pts each)
2. (6pts) Do 2.1.44.
3. (12pts each) Do 2.2.7, 2.2.8, 2.2.9 (4pts each)
4. (6pts) Let T1 , T2 : Rn → Rm be functions that satisfy linear conditions. Suppose further that T1 (ei ) = T2 (ei ) for all standard vectors ei in Rn . Show that
T1 (v) = T2 (v) for all vectors v ∈ Rn .
5. (20pts) Do 2.2.13, 2.2.17, 2.2.15, 2.2.19, 2.2.23 (5pts each).
6. (8pts) The trace of an n × n matrix M , denoted tr M is defined to be the sum
of all the diagonal entries. In other words,
tr M =
n
X
Mii .
i=1
Let A and B be n × n matrices. Prove that
tr (AB) = tr (BA).
7. (5pts) Let A be an n×n matrix. Let u be a row vector in Rn with the i-th entry
being 1 and zero everywhere else. Let v be a column vector in Rn with the
j-th entry being 1 and zero everywhere else. Compute the following matrix
multiplication
uAv
in terms of the entries of A.
8. (8pts) Do 2.3.29, 2.3.66 (4pts each)
9. (8pts) Do 2.3.30.
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