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Problem 1. (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 matrices together z = x × y where x ∈ X and y ∈ Y . Now with this action, let x = 2 1 2 0 2 2 left act on y = 2 1 1 0 0 2 1 0 0 2 , what is the result of action? is the result lies in the same set of y? Is the expression xxy make sense? (2) The same as previous one, but no with X = Y = Z = M 2×2(R), action is also defined by matrix multiplication. now if x = w 1 2 3 1 , y = 1 2 1 , with previous definition, what 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) 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 R 2 (means the element of Z is a subset of R 2 ), 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 R 2 as pictures.(R 2 means set of all points in plane) 2 Define X = { , }, the set of two elements.where each of the operator means the following: 1 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: × + = + = (remember now the addition ”+” in Z is defeined to be taken union sets) (1) Calculate the matrix product (Hint: + = ) (2) Calculate the matrix product (3) Calculate the matrix product (4) Calculate the matrix product 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.)

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Linear Algebra Questions
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