document.write( "Question 61084This question is from textbook Introduction to Linear Algebra
\n" ); document.write( ": I need to find the basis for the null space, the range of the matrix, and the orthogonal basis using Gram-Schmidt.\r
\n" ); document.write( "\n" ); document.write( "The matrix is:\r
\n" ); document.write( "\n" ); document.write( "1 -2 1 -5
\n" ); document.write( "2 1 7 5
\n" ); document.write( "1 -1 2 -2 \n" ); document.write( "
Algebra.Com's Answer #42021 by venugopalramana(3286)\"\" \"About 
You can put this solution on YOUR website!
I WAS WAITING FOR YOUR RESPONSE..ANY WAY I MADE MY OWN ASSUMPTIONS OF THE REQUIREMENTS AND WORKED OUT THE SOLUTION BELOW
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\n" ); document.write( "I need to find the basis for the null space,
\n" ); document.write( "NULL SPACE? WHY?ARE YOU SURE?
\n" ); document.write( " the range of the matrix,
\n" ); document.write( "RANGE? ARE YOU SURE? OR IS IT RANK?
\n" ); document.write( "PLEASE CONFIRM REQUIREMENTS AS THEY LOOK DOUBTFULL!
\n" ); document.write( " and the orthogonal basis using Gram-Schmidt.
\n" ); document.write( "SOLUTION:NULL SPACE , RANK ,ORTHONORMAL BASI\r
\n" ); document.write( "\n" ); document.write( "The matrix is:
\n" ); document.write( "1 -2 1 -5
\n" ); document.write( "2 1 7 5
\n" ); document.write( "1 -1 2 -2
\n" ); document.write( "
\n" ); document.write( "V1= 1 -2 1 -5
\n" ); document.write( "V2= 2 1 7 5
\n" ); document.write( "V3= 1 -1 2 -2
\n" ); document.write( " NR2=R2-2*R1 ; NR3=R3-R1
\n" ); document.write( " 1 -2 1 -5
\n" ); document.write( " 0 5 5 15
\n" ); document.write( " 0 1 1 3
\n" ); document.write( " NR2=R2/5
\n" ); document.write( " 1 -2 1 -5
\n" ); document.write( " 0 1 1 3
\n" ); document.write( " 0 1 1 3
\n" ); document.write( " NR3=R3-R2
\n" ); document.write( " 1 -2 1 -5
\n" ); document.write( " 0 1 1 3
\n" ); document.write( " 0 0 0 0
\n" ); document.write( " NR1=R1-R2
\n" ); document.write( " 1 -3 0 -8
\n" ); document.write( " 0 1 1 3
\n" ); document.write( " 0 0 0 0
\n" ); document.write( "
\n" ); document.write( " RANK = 2
\n" ); document.write( "
\n" ); document.write( " CONSIDERING THE SET OF EQNS.
\n" ); document.write( " AX = 0,WE HAVE ONLY 2 INDEPENDENT EQNS.
\n" ); document.write( " X1-2X2+X3-5X4=0…………….1
\n" ); document.write( " X1-X2+2X3-2X4=0……………2
\n" ); document.write( " EQN.2-EQN.1 GIVES
\n" ); document.write( " X2+X3+3X4=0…..3
\n" ); document.write( " HENCE ONE SET OF SOLUTION COULD BE X2=1,X3=2,X4=-1..AND X1=-5…..THAT IS (-5,1,2,-1) = N1
\n" ); document.write( " AND ANOTHER SET OF SOLUTION COULD BE ..X2=2,X3=1,X4=-1 AND X1=-2..THAT IS (-5,2,1,-1)=N2
\n" ); document.write( " N1 AND N2 BEING 2 INDEPENDENT SOLUTIONS WILL FORM THE BASIS FOR NULL SPACE.
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " HENCE THE 3 VECTORS HAVE 2 INDEPENDENT VECTORS.
\n" ); document.write( " HENCE WE CAN CHOSE ANY 2 INDEPENDENT VECTORS TO FORM BASIS
\n" ); document.write( " LET US TAKE V1 AND V3 WHICH ARE INDEPENDENT TO FORM THE BASIS. \r
\n" ); document.write( "\n" ); document.write( " Y1=V1 = (1,-2,1,-5)
\n" ); document.write( " Y3 =V3+AV1…DOT WITH Y1
\n" ); document.write( " Y3.Y1 = V3.Y1+AV1.Y1…….SINCE Y1 AND Y3 ARE ORTHOGONAL,Y1.Y3=0
\n" ); document.write( " A=-(V3.Y1)/(V1.Y1) = -[1*1+(-1)*(-2)+2*1+(-5)*(-2)]/[1*1+(-2)*(-2)+1*1+(-5)*(-5)]=-15/31
\n" ); document.write( " Y3 = (1,-1,2,-2)-(15/31)(1,-2,1,-5) = (1/31)*(16,-1,47,13)
\n" ); document.write( " HENCE THE 2 ORTHOGONAL VECTORS ARE OBTAINED.
\n" ); document.write( " NORMALISING THEM WE GET THE ORTHONORMAL BASIS
\n" ); document.write( " Y1'=(1,-2,1,-5)/SQRT(31) = (1/5.568)(1,-2,1,-5)
\n" ); document.write( " Y3'= [16,-1,47,13]= (1/51.33)(16,-1,47,13)
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