Linear Algebra Homework 5 Instructions: You can either print out the

Linear Algebra Homework 5
YOUR NAME
Instructions: You can either print out the homework and write your answer on it(recommended),
or other ways that clearly shows your work and computations. Write your UNI in the right top
corner of each page. and your name in the left top corner. For computational excercise, circle the
final answer. This homework is due at 4:30pm June, 15th.
Problem 1. Read the notes Page 68 to Page 71 about the idea of actions. Simply speaking, action
is a combination of symbols xy which is defined to represents some result z = xy, where x ∈ X,
y ∈ Y , z ∈ Z. By language of set, an action can abstractly viewed as a map X × Y → Z, where
it maps each pair (x, y) to the result z = xy. If the result lies in the same set of the object, in
other words, Y = Z, then we can repeatedly apply action on that, that means we can calculate the
expression like x1 x2 x3 · · · xn y. If the result object lies in a different space Z 6= Y we can’t apply
again and again because element of X does not act on Z, for example, take x1 , x2 ∈ X, we don’t
know x1 x2 y because x2 y is already lies in the new set Z, as Z 6= Y , we don’t know how X act on
Z, so x1 (x2 y) is not defined. Please do the following exercises to understand these concepts.
(1) Denote the set of m × n matrices over R by Mm×n (R). Consider the case X = M3×2 (R),
Y = M2×5 (R), and Z = M3×5 (R). The action X ×Y → Z is defined by multiplying
 two ma
2 1
trices together z = x × y where x ∈ X and y ∈ Y . Now with this action, let x =  2 0 
2 2
2
1
0
1
0
left act on y =
, what is the result of action? is the result lies
1
0
2
0
2
in the same set of y? Is the expression xxy make sense?
Date: Printed on:June 10, 2015;
1
Linear Algebra Homework 5
Your UNI:
(2) The same as previous one, but now
2×2 (R), action is also defined by
with X= Y =Z = M
1 2
1 2
matrix multiplication. now if x =
,y=
, with previous definition, what
3 1
1
is the result if x left act on y? what is the result if y right act on x? is the expression xxy
make sense now? If it is, calculate that. (Hint: x left act on y is the same as y right act
on x, so the first two question are the same)
2
YOUR NAME
.
.
Your UNI:
Problem 2. Remember our motivation is to try to put everything into matrix and use the language
of matrix to simplify expressions. if we have an action X × Y → Z, it is in general an idea that
we can define the matrix over X, matrix over Y, and matrix over Z, and cook up the matrix
multiplication using the action. But the definition of matrix multiplication also needs addition, .
That means only with addition structures on Z can we define the matrix multiplication. So we
define concept of semi-group. (Z,+) is called an abelian semi-group if
1, There is an operator of addition ”+” defined on this set. and for any a, b ∈ Z, a + b = c is
an element in Z.
2, a + b = b + a
3, a + (b + c) = (a + b) + c
Now we use some examples to give you an idea of this.
1 Define Z to be the set of all subsets of R2 (means the element of Z is a subset of R2 ), define
the operator ”+”: z1 + z2 := z1 ∪ z2 . Show that with this definition, Z became a abaliean
semi-group. In this excercise, we call subset of R2 as pictures.(R2 means set of all points
in plane)
2 Define X = {
following:
1
,
}, the set of two elements.where each of the operator means the
means rotate the picture clockwise by 90 degree
2
means rotate the picture conterclockwise by 90 degree
Define Y is the same set as Z. in other words, pictures. Now we have action X × Y → Z
in the natrual way:
Example:
×
=
Example:
×
+
= +
Z is defeined to be taken union sets)
=
3
(remember now the addition ”+” in
Linear Algebra Homework 5
Your UNI:





 (Hint:
(1) Calculate the matrix product
4
+
=
)
YOUR NAME
Your UNI:


(2) Calculate the matrix product 







(3) Calculate the matrix product
5
Linear Algebra Homework 5
Your UNI:


(4) Calculate the matrix product 

6






YOUR NAME
Your UNI:
Problem 3. With previous understanding, how do you understand the block matrix multiplication
now? Write some words about your feelings and understandings.(In the case of block matrix multiplication, block matrix could be viewed as matrix of which kind of object? and what is the object,
what is the action, where is the structure of semi-group, etc.)
7