Assignment 8–Algebra I Due at the beginning of tutorial Nov. 27 1 ...

Assignment 8–Algebra I
Due at the beginning of tutorial Nov. 27
1. Compute AB where /3
˙1
A =
˙2
˙2
˙2
∈ M22(Z3)
and
B =
˙0 ˙2 ˙1
˙1 ˙2 ˙0
∈ M23(Z3).
Show enough work to convince your marker that you know what you are
doing.
2. Suppose F is a field and A ∈ Mnn(F ) is a diagonal matrix, which means
[A]kj = 0 whenever k = j. Suppose further that [A]kk = [A]jj for k = j,
and B ∈ Mnn(F ).
(a) Compute [BA]kj for all k = j. /2
(b) Compute [AB]kj for all k = j. /2
(c) Prove that AB = BA if and only if B is a diagonal matrix. /6
3. Suppose A ∈ Mnn(C) satisfies 〈Av, Aw〉 = 〈v, w〉 for all v, w ∈ C n . Prove /6
that A is unitary. (Hint: Use exercise 2 (a)-(b) from assignment 7. Be
sure to use the definition of unitary given in the lectures (not the text).)
SUGGESTED EXERCISES
(see next page)
• Exercises C21-C25 in section MISLE.EXC.
• Exercises T10 and T11 in section MINM.EXC.
• Suppose F is a field, α1, . . . , αm ∈ F , and v1, . . . , vm ∈ F n .
induction on m ≥ 1 that
Prove by
⎡ ⎤
m
⎣ ⎦
m
= [vj]k, 1 ≤ k ≤ n
and ⎡
m
⎣
j=1
j=1
vj
αjvj
⎤
⎦
k
k
=
j=1
m
αj[vj]k, 1 ≤ k ≤ n.
j=1
• Suppose F is a field, A ∈ Mmn(F ) and ej ∈ F n is the jth standard basis
vector. Prove
[Aej]k = [A]kj, 1 ≤ j ≤ n, 1 ≤ k ≤ m.
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• Suppose F is a field. We define a matrix A ∈ Mnn(F ) to be upper-triangular
if [A]kj = 0 for all 1 ≤ j < k ≤ n.
1. Give an example of an upper-triangular matrix.
2. Suppose A, B ∈ Mnn(F ) are upper-triangular. Prove that AB is
upper-triangular.
3. Suppose A ∈ Mnn(F ) is upper-triangular. Prove that A m is uppertriangular
by induction on m ≥ 1.
4. Give a definition for a square matrix to be lower-triangular. Do the
previous result remain true for lower-triangular matrices?
• Suppose A ∈ Mnn(C) and v, w ∈ C n . Prove that
by using Theorem AIP.
〈v, Aw〉 = 〈A ∗ v, w〉
• Suppose A ∈ Mnn(C) is invertible. Prove that
and
A −1 = ( Ā)−1
(A ∗ ) −1 = (A −1 ) ∗ .
• Give examples of invertible matrices A and B such that A + B is not
invertible.
• Give examples of invertible matrices A and B such that A+B is invertible,
but (A + B) −1 = A −1 + B −1 .
• Give examples of non-zero matrices A and B such that AB is the zero
matrix.
• Pick a matrix in M22(Z5) and determine whether it is invertible or not. If
it is invertible, compute its inverse. Repeat this exercise until comfortable.
Try with Z7 in place of Z5.
• Suppose {v1, . . . , vk} ⊂ C n is an orthonormal set and w ∈ span{v1, . . . , vk}.
Prove that w = k
j=1 〈w, vj〉vj. This gives us a simple way of computing
a linear combination expressing w!
• Suppose F is a field and A, B ∈ Mmn(F ). Prove that A = B if and only if
vA = vB for all row vectors v ∈ F m . (Hint: Imitate Theorem EMMVP.
For this exercise think of v as a 1 × m matrix.)
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