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July 10th, 2010, 12:57 AM   #1
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vector spaces

Let V be a set of all 2x2 matrices such that abcd= 0.

a) is V closed under addition: The answer is No.
My work: (I don't know if this is right )

and


so
(a+e)(b+f)(c+g)(d+h)=(ab+af+eb+ef)(cd+ch+gd+gh)=0
It does not satisfy the constraints abcd=0 so it does not closed under addition.

[color=#BF0000]b)[/color] is V closed under scalar multiplication? The answer is Yes.



c) what is the zero vector in the set V



d)Does every matrix A in V have a negative that is in V? explain. The answer is Yes.


So -A in V, since (-a)(-b)(-c)(-d)=0

e)Is V a vector space? Explain. '
The answer is No because V is not under closed multiplication (But I have no idea why it says that when the answer to [color=#FF0000](b)[/color] is "yes )

I know I did not show my work correctly. I somewhat understand how to determine if V is a vector space for a 2x1 matrices but I am a bit confused finding the vector space for a 2x2 matrices. Can someone please help me out. Thanks in advance
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July 10th, 2010, 04:20 AM   #2
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Re: vector spaces

Your notation is a little strange; it isn't usual to use when manipulating matrices in the normal way. Is that what your textbook does?

First of all, abcd = 0 if and only if at least one of {a,b,c,d} is zero.

(a)

No, and give the counterexample of . Nothing more is needed.

(b)

Yes; if one of {a,b,c,d} is zero, then one of {ra,rb,rc,rd} is zero.

(c)

Your answer is correct.

(d)

You must mean surely? Remember to point out that if one of {a,b,c,d} is zero, one of {-a,-b,-c,-d} is zero.

(e)

No because the requirements are not fulfilled, (a) being one of the requirements. When it says that V is not closed under multiplication it must mean matrix multiplication.
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