document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=62805>Question 62805</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi,my name is Natalia. I solved two problems, but I'm not sure that I did it right. I would really appreciate if you can check them.<BR>\n" );
document.write( "Here are two given problem:<BR>\n" );
document.write( "Consider a vector space V=Mat 3x3 of all 3x3 matrices<BR>\n" );
document.write( "Q.: determine the dimension of V,find and verify a basis for it.\r<BR>\n" );
document.write( "\n" );
document.write( "here is what i've got so far:\r<BR>\n" );
document.write( "\n" );
document.write( "if the matrix is <BR>\n" );
document.write( "/a b c/  <BR>\n" );
document.write( "/d e f/       <BR>\n" );
document.write( "/g h i/<BR>\n" );
document.write( "then the basis for it is<BR>\n" );
document.write( "/1 0 0/ /0 1 0/ /0 0 1/ /0 0 0/ /0 0 0/ /0 0 0/ /0 0 0/ /0 0 0/ /0 0 0/<BR>\n" );
document.write( "/0 0 0/,/0 0 0/,/0 0 0/,/1 0 0/,/0 1 0/,/0 0 1/,/0 0 0/,/0 0 0/,/0 0 0/<BR>\n" );
document.write( "/0 0 0/ /0 0 0/ /0 0 0/ /0 0 0/ /0 0 0/ /0 0 0/ /1 0 0/ /0 1 0/ /0 0 1/         \r<BR>\n" );
document.write( "\n" );
document.write( "So, i have 9 vectors v1, v2,...,v9 that are lineraly independent and span<BR>\n" );
document.write( "therefore, dimV=9 if the matrix is symmetric dimV=6 and if the matrix is scew-symmetric dimV=9  Am I correct?\r<BR>\n" );
document.write( "\n" );
document.write( "And the second problem:\r<BR>\n" );
document.write( "\n" );
document.write( "A have a matrix A=<BR>\n" );
document.write( "/1 2 3 /                                                      <BR>\n" );
document.write( "/2 5 7 /<BR>\n" );
document.write( "/3 7 10/               <BR>\n" );
document.write( "that lookes like A=<BR>\n" );
document.write( "/1 0 1/  <BR>\n" );
document.write( "/0 1 1/        <BR>\n" );
document.write( "/0 0 0/ when row-reduced<BR>\n" );
document.write( "                   <BR>\n" );
document.write( "I need to express a vector x=<BR>\n" );
document.write( "/2 /  <BR>\n" );
document.write( "/-1/   <BR>\n" );
document.write( "/7 /                               <BR>\n" );
document.write( "as a sum of a vector from the row space and a vector from nullspace<BR>\n" );
document.write( "For nullspace I got (let's call it v so it will be easier to write an expression then)  <BR>\n" );
document.write( "/-1/           <BR>\n" );
document.write( "/-1/                       <BR>\n" );
document.write( "/1 /                       <BR>\n" );
document.write( "for row space (u and w respectively)<BR>\n" );
document.write( "/1/ /0/<BR>\n" );
document.write( "/0/ /1/<BR>\n" );
document.write( "/1/ /1/<BR>\n" );
document.write( "that gives me 3-D subspace<BR>\n" );
document.write( "and the vector expression is:<BR>\n" );
document.write( "x=2*v + 4*u + 1*w<BR>\n" );
document.write( "did I solve it correct?\r<BR>\n" );
document.write( "\n" );
document.write( "Thank you very much! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #43646 by </B><A HREF=/tutors/aboutme.mpl?userid=venugopalramana><B>venugopalramana(3286)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=venugopalramana><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.geocities.com/venugopalramana/FREE_MATHS_ASSISTANCE.html valign=middle><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=43646\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> GOOD TO SEE YOUR SINCERE ATTEMPT TO LEARN,DO AND THEN ASK FOR VERIFICATION.<BR>\n" );
document.write( "KEEP IT UP!IT WILL TAKE YOU TO THE TOP IF YOU PURSUE THIS METHOD!<BR>\n" );
document.write( "AT THE OUTSET YOU HAVE DONE LET ME SAY THAT YOU HAVE DONE PERFECTLY WELL<BR>\n" );
document.write( "SEE MY COMMENTS BELOW.I AM GIVING ADDITIONAL SUGGESTIONS FOR IMPROVEMENT <BR>\n" );
document.write( "IN BRACKETS WHICH YOU MAY OMIT IN FIRST READING.<BR>\n" );
document.write( "Hi,my name is Natalia. I solved two problems, but I'm not sure that I did it right. I would really appreciate if you can check them.<BR>\n" );
document.write( "Here are two given problem:<BR>\n" );
document.write( "Consider a vector space V=Mat 3x3 of all 3x3 matrices<BR>\n" );
document.write( "Q.: determine the dimension of V,find and verify a basis for it.<BR>\n" );
document.write( "here is what i've got so far:<BR>\n" );
document.write( "if the matrix is<BR>\n" );
document.write( "/a b c/<BR>\n" );
document.write( "/d e f/<BR>\n" );
document.write( "/g h i/OK....[IF YOU ARE DEALING WITH SMALLER MATRICES..YOU BETTER GET <BR>\n" );
document.write( "ACCUSTOMED TO PUT IT AS<BR>\n" );
document.write( "A1,A2,A3<BR>\n" );
document.write( "B1,B2,B3<BR>\n" );
document.write( "C1,C2,C3....WHERE COLUMNS ARE DESIGNATED BY 1,2,3 AND ROWS BY A,B,C, ETC<BR>\n" );
document.write( "OR BETTER STILL<BR>\n" );
document.write( "A11,A12,A13<BR>\n" );
document.write( "A21,A22,A23<BR>\n" );
document.write( "A31,A32,A33..WHERE ROWS ARE DENOTED BY I DIGIT AND COLUMNS BY SECOND DIGIT IN THE SUFFIX..KEEPING A SAME FOR THIS MATRIX WHICH WE CAN CALL AS MATRIX A. THEN WE CAN CALL ANOTHER MATRIX B BY FOLLOWING SAME TERMINOLOGY.B11,B12..ETC]\r<BR>\n" );
document.write( "\n" );
document.write( "then the basis for it is....<BR>\n" );
document.write( "-----------------------------------------------------<BR>\n" );
document.write( "HERE I THINK WE BETTER REFLECT CLEARLY ABOUT 'BASIS'<BR>\n" );
document.write( "LET US DISTINGUISH 3 THINGS FIRST.<BR>\n" );
document.write( "1.SET OR OUR UNIVERSE:- IN OUR SET OR UNIVERSE UNDER CONSIDERATION ,<BR>\n" );
document.write( " THERE COULD BE 1 OR 2 OR 3...<BR>\n" );
document.write( "OR N MATRICES IF IT IS A FINITE SET OR INFINITE MATRICES IF IT IS <BR>\n" );
document.write( "INFINITE SET AS IN YOUR<BR>\n" );
document.write( "PROBLEM NOW..THERE ARE INFINITE NUMBER OF 3x3 MATRICE.IF IT IS <BR>\n" );
document.write( "FINITE AND THERE ARE N MATRICES WE SAY OUR UNIVERSE IS A SET OF N GIVEN MATRICES.<BR>\n" );
document.write( "2.DIMENSION:- THIS IS THE MINIMUM NUMBER OF LINEARLY INDEPENDENT MATRICES<BR>\n" );
document.write( " SAY R MATRICES NEEDED TO BE TAKEN FROM THE ABOVE UNIVERSE TO EXPRESS<BR>\n" );
document.write( " EVERY MATRIX IN THE UNIVERSE AS A LINEAR COMBINATION OF THESE R <BR>\n" );
document.write( " MATRICES.R<=N ALWAYS. <BR>\n" );
document.write( "3.BASIS:-YOU WILL FIND THAT THE ABOVE R MATRICES ARE NOT UNIQUE<BR>\n" );
document.write( " IN GENERAL, UNLESS R=N, IN THE <BR>\n" );
document.write( " SENSE THAT ANY SET OF R LINEARLY INDEPENDENT MATRICES FROM THE UNIVERSE<BR>\n" );
document.write( " COULD BE PICKED UP TO EXPRESS EVERY MATRIX IN THE UNIVERSE AS A LINEAR <BR>\n" );
document.write( " COMBINATION OF THOSE R MATRICES.ANY SUCH SET OF R MATRICES , WE CALL<BR>\n" );
document.write( " A BASIS FOR THE UNIVERSE.THERE COULD BE BASIS B1,BASIS B2 ETC..<BR>\n" );
document.write( " BUT ALL OF THEM WILL HAVE R AND ONLY R NUMBER OF LINEARLY <BR>\n" );
document.write( " INDEPENDENT MATRICES.OK.?<BR>\n" );
document.write( " ELEMENTARY BASIS:- THE EASIEST BASIS TO CONCEIVE IS (1,0,0,0....ETC),<BR>\n" );
document.write( "(0,1,0,0...ETC),(0,0,1,0...ETC)....ETC...DEPENDING ON THE DIMENSION.<BR>\n" );
document.write( "FOR 3D , WE WILL HAVE 3 , FOR 4D...4 ETC..<BR>\n" );
document.write( "------------------------------------------------  <BR>\n" );
document.write( "NOW WE ARE TRYING TO PUT UP A BASIS FOR A UNIVERSE OF INFINITE SET <BR>\n" );
document.write( "OF MATRICES.BUT FIRST LET US BE CLEAR ABOUT ITS DIMENSION.SINCE IT<BR>\n" );
document.write( " IS A UNIVERSE OF 3X3 MATRICES , EACH OF WHICH CANHAVE 3*3=9 <BR>\n" );
document.write( " ELEMENTS,ALL INDEPENDENT IN THE SENSE THAT THEY CAN TAKE ANY<BR>\n" );
document.write( "VALUE FROM THE REAL FIELD, THE DIMENSION WE CAN GUESS TO BE 3*3=9.[IN <BR>\n" );
document.write( "NERAL FOE A NXN MATRIX WE HAVE DIMENSION OF N^2<BR>\n" );
document.write( " NEXT IS TO PUT UP A....STANDARD E BASIS.<BR>\n" );
document.write( " THIS IS EASILY DONE BY 9 MATRICES M1,M2,M3...M9 AS FOLLOWS. <BR>\n" );
document.write( "M1=<BR>\n" );
document.write( "1,0,0<BR>\n" );
document.write( "0,0,0<BR>\n" );
document.write( "0,0,0<BR>\n" );
document.write( "-----------------<BR>\n" );
document.write( "M2=<BR>\n" );
document.write( "0,1,0<BR>\n" );
document.write( "0,0,0<BR>\n" );
document.write( "0,0,0<BR>\n" );
document.write( "-------------<BR>\n" );
document.write( "M3=<BR>\n" );
document.write( "0,0,1<BR>\n" );
document.write( "0,0,0<BR>\n" );
document.write( "0,0,0<BR>\n" );
document.write( "----------------<BR>\n" );
document.write( "M4=<BR>\n" );
document.write( "0,0,0<BR>\n" );
document.write( "1,0,0<BR>\n" );
document.write( "0,0,0<BR>\n" );
document.write( "----------------ETC..AS IN SUCH A CASE ANY MATRIX 'A' CAN BE GOT BY  <BR>\n" );
document.write( "A11*M1+A12*M2+....ETC....A33*M9....WHERE A IS MATRIX EXPLAINED IN <BR>\n" );
document.write( "THE BEGINING AS YOU HAVE DONE CORRECTLY <BR>\n" );
document.write( "/1 0 0/ /0 1 0/ /0 0 1/ /0 0 0/ /0 0 0/ /0 0 0/ /0 0 0/ /0 0 0/ /0 0 0/<BR>\n" );
document.write( "/0 0 0/,/0 0 0/,/0 0 0/,/1 0 0/,/0 1 0/,/0 0 1/,/0 0 0/,/0 0 0/,/0 0 0/<BR>\n" );
document.write( "/0 0 0/ /0 0 0/ /0 0 0/ /0 0 0/ /0 0 0/ /0 0 0/ /1 0 0/ /0 1 0/ /0 0 1/<BR>\n" );
document.write( "So, i have 9 vectors v1, v2,...,v9 that are lineraly independent and span<BR>\n" );
document.write( "therefore, dimV=9<BR>\n" );
document.write( "PERFECT!!\r<BR>\n" );
document.write( "\n" );
document.write( " if the matrix is symmetric dimV=6<BR>\n" );
document.write( "INDEPENDENT ELEMENTS WE HAVE HERE ARE 1+2+3....N IN I,II,....NTH.ROW<BR>\n" );
document.write( "=N(N+1)/2 IN GENERAL ...HERE 3*4/2=6  <BR>\n" );
document.write( "EXCELLENT<BR>\n" );
document.write( "and if the matrix is skew-symmetric dimV=9 Am I correct?<BR>\n" );
document.write( " NO !DEAR!YOU MADE A SLIP HERE.<BR>\n" );
document.write( " FROM THE ABOVE ANSWER,WE HAVE TO SUBTRACT N DIAGONAL ELEMENTS AS <BR>\n" );
document.write( " THEY ARE ALL ZEROS.SO WE GET N(N+1)/2 - N = N(N-1)/2 ..<BR>\n" );
document.write( " THAT IS 3*2/2=3 IS THE ANSWER.<BR>\n" );
document.write( "And the second problem: I SHALL ANSWER LATER.LET ME POST THIS<BR>\n" );
document.write( " FIRST FOR YOU.<BR>\n" );
document.write( "==============================================<BR>\n" );
document.write( "A have a matrix A=<BR>\n" );
document.write( "/1 2 3 /<BR>\n" );
document.write( "/2 5 7 /<BR>\n" );
document.write( "/3 7 10/OK<BR>\n" );
document.write( "that lookes like A=<BR>\n" );
document.write( "/1 0 1/<BR>\n" );
document.write( "/0 1 1/<BR>\n" );
document.write( "/0 0 0/ when row-reduced<BR>\n" );
document.write( "VERY GOOD<BR>\n" );
document.write( "I need to express a vector x=<BR>\n" );
document.write( "/2 /<BR>\n" );
document.write( "/-1/<BR>\n" );
document.write( "/7 /OK<BR>\n" );
document.write( "as a sum of a vector from the row space and a vector from nullspace<BR>\n" );
document.write( "For nullspace I got (let's call it v so it will be easier to write an expression then)<BR>\n" );
document.write( "/-1/<BR>\n" );
document.write( "/-1/<BR>\n" );
document.write( "/1 /<BR>\n" );
document.write( "EXCELLENT<BR>\n" );
document.write( "for row space (u and w respectively)<BR>\n" );
document.write( "/1/ /0/<BR>\n" );
document.write( "/0/ /1/<BR>\n" );
document.write( "/1/ /1/<BR>\n" );
document.write( "VERY GOOD<BR>\n" );
document.write( "that gives me 3-D subspace<BR>\n" );
document.write( "and the vector expression is:<BR>\n" );
document.write( "x=2*v + 4*u + 1*w<BR>\n" );
document.write( "EXCELLENT !!!KEEP IT UP.YOU ARE REALLY GOOD!ONLY YOU NEED A<BR>\n" );
document.write( " LITTLE SELF CONFIDENCE. BELIEVE IN YOUR ABILITY!GREAT JOB!!<BR>\n" );
document.write( "did I solve it correct?<BR>\n" );
document.write( "Thank you very much! </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );