July 10th, 2010, 12:57 AM  #1 
Member Joined: Feb 2009 Posts: 76 Thanks: 0  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 
July 10th, 2010, 04:20 AM  #2 
Senior Member Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3  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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