document.write( "Question 26364: Consider the accompanying matrix. Use the test for linear independence to find a basis for the space spanned by the rows of the matrix. Suppose that this matrix is augmented matrix for a system of equations. What is the rank of this systen? Which equations can be discarded?\r
\n" ); document.write( "\n" ); document.write( "{[1 0 1 1
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\n" ); document.write( "3 3 6 -3
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Algebra.Com's Answer #14289 by venugopalramana(3286)\"\" \"About 
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Consider the accompanying matrix. Use the test for linear independence to find a basis for the space spanned by the rows of the matrix. Suppose that this matrix is augmented matrix for a system of equations. What is the rank of this systen? Which equations can be discarded?
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\n" ); document.write( "THE SECOND QUESTION FIRST ...AS PER THE SECOND QUESTION ,THIS IS AN AUGMENTED MATRIX.HENCE LAST COLUMN IS CONSTANTS COLUMN.
\n" ); document.write( "AND THE FIRST 3 COLUMNS ARE COEFFICIENT MATRIX.BUT THERE ARE 4 ROWS.THAT IS THERE ARE 3 UNKNOWNS AND 4 EQNS.LET US FIND RANK OF ASUGMENTED MATRIX...
\n" ); document.write( " 1 0 1 1
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\n" ); document.write( "\n" ); document.write( "R1=R1………… 1 0 1 1
\n" ); document.write( "R2=R2-2R1….. 0 1 1 -2
\n" ); document.write( "R3=R3-3R1….. 0 3 3 -6
\n" ); document.write( "R4=R4-4R1…… 0 1 1 -2 \r
\n" ); document.write( "\n" ); document.write( "R1=R1………… 1 0 1 1
\n" ); document.write( "R2=R2…….….. 0 1 1 -2
\n" ); document.write( "R3=R3-3R2….. 0 0 0 0
\n" ); document.write( "R4=R4-4R1…… 0 0 0 0
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\n" ); document.write( "HENCE RANK = 2
\n" ); document.write( "EQNS.2 AND 4 ARE LEADING TO SAME RESULT.
\n" ); document.write( "AND EQN 3 AND 4 ARE ALL ZEROES . SO WE CAN DISCARD EQNS.3 AND 4 IN THIS SYSTEM.
\n" ); document.write( "SO WE REALLY HAVE 2 INDEPENDENT EQNS. IN 3 UNKNOWNS LEADING TO INFINITE SOLUTIONS.
\n" ); document.write( "NOW COMING TO YOUR FIRST QUESTION ,THE DIMENSIONS OF THE BASIS IS NOT GIVEN.
\n" ); document.write( "TAKING 4 DIMENSIONAL BASIS FOR 4 EQNS.,WE GET
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\n" ); document.write( "EQN.1 …..R1= A+C+D
\n" ); document.write( "EQN2…….R2= 2A+B+3C
\n" ); document.write( "EQN.3……R3= 3A+3B+6C-3D
\n" ); document.write( "EQN.4……R4= 4A+B+5C+2D
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\n" ); document.write( "BUT WE GOT R4-4R1=B+C-2D=R2-2R1……..OR……….R4=R2+2R1
\n" ); document.write( "AND…………..R3-3R2+3R1=0………………….OR……….R3=3R2-3R1
\n" ); document.write( "THAT IS TAKING R1 AND R2 AS 2 INDEPENDENT EQNS. WE SHOWED R3 AND R4 AS A LINEAR COMBINATION OF
\n" ); document.write( "R1 AND R2.
\n" ); document.write( "HENCE THE BASIS FOR THESE SET OF 4 EQNS.CAN BE TAKEN AS
\n" ); document.write( "R1=A+C+D....AND.....R2=2A+B+3C
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