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 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 $\ A= \begin{bmatrix} a &b \\ c & d \end{bmatrix}\$ such that abcd= 0. a) is V closed under addition: The answer is No. My work: (I don't know if this is right ) $\ u\oplus v=v\oplus u\$ $\ u=\begin{bmatrix} a &b \\ c& d \end{bmatrix}\$ and $\ v=\begin{bmatrix} e &f \\ g& h \end{bmatrix}\$ $\ \begin{bmatrix} a &b \\ c& d \end{bmatrix}+\begin{bmatrix} e &f \\ g &h \end{bmatrix}=\begin{bmatrix} a+e & b+f\\ c+g &d+h \end{bmatrix}\$ 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\odot (u\oplus v)=c\odot u\oplus c\odot v\$ $\ r\cdot (\begin{bmatrix} a & b\\ c & d \end{bmatrix}+\begin{bmatrix} e &f \\ g&h \end{bmatrix})=\begin{bmatrix} ra & rb\\ rc& rd \end{bmatrix}+\begin{bmatrix} re&rf \\ rg& rh \end{bmatrix}=(ra\, rb)+(rc\, rd) +?????\,,Don't\,know\$ c) what is the zero vector in the set V $\ u\oplus O=O\oplus u=u\$ $\ \begin{bmatrix} a & b\\ c & d \end{bmatrix}\oplus \begin{bmatrix} 0 &0 \\ 0 &0 \end{bmatrix}=\begin{bmatrix} a&b \\ c &d \end{bmatrix}\$ d)Does every matrix A in V have a negative that is in V? explain. The answer is Yes. $\ u\oplus -u=-u\oplus u=0\$ $\ \begin{bmatrix} a & b\\ c&d \end{bmatrix}+\begin{bmatrix} -a &b \\ c &d \end{bmatrix}=\begin{bmatrix} 0&0 \\ 0&0 \end{bmatrix}\$ 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 $\oplus \odot$ 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 $\begin{bmatrix}1&0\\1&0\end{bmatrix}+\begin{bmatri x}0&1\\0&1\end{bmatrix}$. 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 $\begin{bmatrix}-a&-b\\-c&-d\end{bmatrix}$ 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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# show that the set of all 2x2 matrices are closed under multiplication

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