document.write( "Question 22330: Let v1=(1,6,4), v2=(2,4,-1), v3=(-1,2,5) and w1=(1,-2,-5), w2=(0,8,9). Prove that span(v1,v2,v3)=span(w1,w2). thank you \n" ); document.write( "
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v1=(1,6,4), v2=(2,4,-1), v3=(-1,2,5) and w1=(1,-2,-5), w2=(0,8,9). Prove that span(v1,v2,v3)=span(w1,w2). \r
\n" ); document.write( "\n" ); document.write( " Try to solve
\n" ); document.write( " aw1 + bw2 = v1,\r
\n" ); document.write( "\n" ); document.write( " cw1 + dw2 = v2,
\n" ); document.write( "and
\n" ); document.write( " ew1 + fw2 = v3.\r
\n" ); document.write( "\n" ); document.write( " i.e. a(1,-2,-5)+b(0,8,9)=( 1,6,4) ...(1)
\n" ); document.write( " c(1,-2,-5)+d(0,8,9)=(2,4,-1)...(2)
\n" ); document.write( " e(1,-2,-5)+ f(0,8,9)=(-1,2,5)...(3)\r
\n" ); document.write( "\n" ); document.write( " (1) becomes:
\n" ); document.write( " a+0=1, -2a+8b = 6, -5a+9b = 4.
\n" ); document.write( "So,a=-1, b=1.\r
\n" ); document.write( "\n" ); document.write( " Solving (b),(c) left for you to see if they have feasible
\n" ); document.write( " solutions. so, then v1,v2,v3 are in the span {w1,w2}.
\n" ); document.write( "
\n" ); document.write( " Next. since w1, w2 are indep, dim spn(w1,w2} =2.
\n" ); document.write( " Also, dim span(v1,v2,v3) >=2.
\n" ); document.write( " Hence, span(v1,v2,v3)=span(w1,w2) \r
\n" ); document.write( "\n" ); document.write( " Kenny \n" ); document.write( "
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