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-- Tutorial 2: Vector Spaces and Modules                            --
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-- Use C-x 2 to get a horizontally split window or 
-- Use C-x 3 to get a vertically split window.

-- Select the lower (or right) window and Hit F12 to start M2.

-- Hit F11 to send code from this screen to M2.

-- for help with M2 type (works in version 1.1)

viewHelp

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-- Free Modules                                                     --
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-- How do you create the free ZZ-module with 1 generator?



-- How do you create the free ZZ-module with 16 generators?



-- Let A be your favorite ring.  How do you create the free A-module
-- with 12 generators?



-- Explain what the following code does:

A = ZZ;
f = matrix{{1,2,3},{4,5,6}}
source f
target f



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-- Matrices                                                        --
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-- Why is it natural to study matrices when studying free modules?



-- Consider the following ring and matrix:

restart
A = ZZ[a..i]
phi = genericMatrix(A,3,3)

-- How do you easily create the matrix below?
--
--    | a b c |
--    | d e f | 
--    | g h i |



-- How do you extract a given row of a given matrix?  Give an example.



-- How do you extract a given column of a given matrix?  Give an
-- example.



-- How you extract a given entry of a given matrix?  Give an example.



-- Explain what the following code does:

A = ZZ[a..i];
phi = genericMatrix(A,3,3)
# phi



-- Bart is typing the following code:

A = ZZ[a..i];
f = genericMatrix(A,3,3)

-- Now Bart wants to make a bigger ring, say R:

R = ZZ[a..z];

-- Why is Bart having problems?  Help Bart fix his code.



-- Consider the ring A = ZZ/2[x,y,z]. How do you create the matrix 
--
--    | x 0 y |
--    | 0 y z | with entries in A?
--    | x 0 0 |




-- Consider the ring A = ZZ/2[x,y,z]. How do you create the matrix 
--
--    | 1 0 1 |
--    | 0 1 1 | with entries in A?
--    | 1 0 0 |



-- Consider A = ZZ/5051[x,y,z].  Here is the module A^1 mod the ideal
-- (x^2).

A = ZZ/5051[x,y,z]
A^1/ideal(x^2)

-- Explain what Macaulay 2 is telling you.



----------------------------------------------------------------------
-- Modules and Ideals                                               --
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-- Every ideal of a ring A is an A-module.  However, Macaulay 2 needs
-- to be told if you wish to think of a ideal as a module.  See if you
-- can get Macaulay 2 to think of the following ideal as a module.

A = ZZ/101[x,y,z]
I = ideal(x^3,y^4*z)



-- Given a ring A, an A-module is an ideal if it is a subset of A.
-- See if you can get Macaulay 2 to think of the following module as a
-- ideal.

A = ZZ/101[x,y,z]
f = matrix{{x^3,y^4*z}}
M = image f



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-- FIN                                                              --
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