Re: Help computing a base for submodules (corrected)?

themadhatter012@xxxxxxxxx wrote:
I need to compute the base for submodules but my only experience is
with vector spaces and I'm a little fuzzy even there. All of the
modules are over a PID.

1. Find a base for the submodule of Z^(3) (Z^(3) is a module of rank 3
over Z -- the integers) generated by f1=(1,0,-1), f2=(2,-3,1),
f3=(0,3,1) and f4=(3,1,5).

Here I constructed a matrix with the 4 generators

|1 0 (-1)|
|2 (-3) 1|
||0 3 1|
|3 1 5|

and via row operations I reduced this to

|1 0 0|
|0 1 0|
|0 0 1|
|0 0 0|

So a basis would consist of {100, 010, 001} which generates all of
Z^(3), right?

Assuming that your arithmetic is correct ( I think that it is).
A simple check is to ensure that each of the original generators can be
written in terms of your basis (trivial here) and that each of the
basis vectors can be written in terms of the original generators (I
trust that it can be done)


2. Find a base for the submodule of Q[x]^(3) [here Q[x]^(3) is a
module of rank 3 over Q[x], the polynomial ring with rational
coefficients] generated by f1=(2x-1,x,(x^2) + 3), f2=(x,x,(x^2)), and
f3=(x+1,2x,(2x^2) -3).

By row operations here (I belive I did this correctly) I went from

|2x-1 x (x^2) + 3 |
|x x x^2 |

|x+1 2x (2x^2) -3|

to

|1 x (x ^2)-3|
|0 0 0 |
|0 0 0 |

Here I'm not so sure but would the basis then consist of just {1
x (x ^2)-3}?

This is wrong. You can see that none of the original generators lie in
the module generated by [ 1, x, x^2 - 3 ]

I believe that a basis is the two vectors
[ 2x - 1 , x , x^2 + 3 ] and [ 1 - x , 0 , -3 ]
Assuming that I did the algebra correctly.



3. Find a base for the Z-submodule of Z^(3) consisting of all
(x1,x2,x3) satisfying the conditions x1+2x1+3x3 = 0 and x1+4x2+9x3=0.

Finally, here we have x1 + 2x1 +3x3 = 3x1 +3x3 = 0 and x1+4x2+9x3=0.

So we get x3 = -x1 and x2 = 2x1 by substitution and the fact that we
are in an integral domain.

Ok. ( are you sure that x1+2x1+3x3 isn't actually x1+2x2+3x3 with a
copying error.)

Everything can be expressed in terms of x1,
so the dimension of the basis should just be one. Now I'm not
really sure what a basis would consist of. I would appreciate any
help.
(1,2,-1)


TMH

.

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