document.write( "Question 1181386: Let b1, b2, b3 be 3 vectors over Z5 with dimensions 3x1.
\n" ); document.write( "b1, b2, b3 are linearly independent.
\n" ); document.write( "A is a matrix over Z5 with dimensions 3x3.
\n" ); document.write( "Assuming that B is a matrix with dimensions 3x3 where it’s columns are b1 b2 b3.
\n" ); document.write( "If it’s given that:
\n" ); document.write( "1 2 4
\n" ); document.write( "3 1 2 = A*B
\n" ); document.write( "1 2 4\r
\n" ); document.write( "\n" ); document.write( "Find a vector basis for the vector space that is the answers for the equation Ax=0. | x’s dimensions are 3x1.
\n" ); document.write( "(The answer may include elements of bi (you can define them however you’d like)) \n" ); document.write( "

Algebra.Com's Answer #811287 by ikleyn(52879) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "Let b1, b2, b3 be 3 vectors over Z5 with dimensions 3x1.
\n" ); document.write( "b1, b2, b3 are linearly independent.
\n" ); document.write( "A is a matrix over Z5 with dimensions 3x3.
\n" ); document.write( "Assuming that B is a matrix with dimensions 3x3 where it’s columns are b1 b2 b3.
\n" ); document.write( "If it’s given that:
\n" ); document.write( "1 2 4
\n" ); document.write( "3 1 2 = A*B
\n" ); document.write( "1 2 4\r
\n" ); document.write( "\n" ); document.write( "Find a vector basis for the vector space that is the answers for the equation Ax=0. | x’s dimensions are 3x1.
\n" ); document.write( "(The answer may include elements of bi (you can define them however you’d like))
\n" ); document.write( "~~~~~~~~~~~~~~~\r
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\n" ); document.write( "\n" ); document.write( "

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document.write( "If matrix B is comprised of vectors b1, b2 and b3 as the columns, then the columns of the matrix A*B are :\r\n" );
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document.write( "    1st column of  A*B  is the vector  Ab1\r\n" );
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document.write( "    2nd column of  A*B  is the vector  Ab2,  and\r\n" );
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document.write( "    3rd column of  A*B  is the vector  Ab3.\r\n" );
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document.write( "Next, notice that\r\n" );
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document.write( "    (1)  in matrix A*B  the third column is twice its second column;  it means that  Ab3 = 2Ab2;\r\n" );
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document.write( "    (2)  in matrix A*B  the sum of the first column and the third column is the vector  ,  which over the field Z5 is the same as   \r\n" );
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document.write( "                   it means that  Ab1 + Ab3 is the zero vector in this 3D space  ;  in other words, Ab1 + Ab3 = 0.\r\n" );
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document.write( "Notices (1) and (2) combined  MEAN  that Matrix A has the kernel of the dimension at least 2 over the field ;\r\n" );
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document.write( "at the same time, the kernel dimension IS NOT 3, as it is seen from the matrix A*B  (! it is not a zero matrix ! ) - - - hence, \r\n" );
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document.write( "the dimension of the kernel is EXACTLY 2.\r\n" );
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document.write( "Now, collecting all our observations, we can conclude that the kernel of A has dimension of 2 over the field  , \r\n" );
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document.write( "and the basis vectors of the kernel are  2b2- b3  and  b1 + b3  ( that are linearly independent !)\r\n" );
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\n" ); document.write( "\n" ); document.write( "At this point, the solution is completed.\r
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\n" ); document.write( "\n" ); document.write( "This problem is intended for students who understand this my explanation and for whom this explanation is enough.\r
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\n" ); document.write( "\n" ); document.write( "Please do not forget to post your \"THANKS\" to me for my teaching (!)\r
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\n" ); document.write( "\n" ); document.write( "From which University / thinking center / Abstract ALGEBRA textbook / (problems book) is this problem ?\r
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