document.write( "Question 839172: Let W be a subset of R3 be the subspace spanned by the vectors w1 and w2 below. Find the coordinates of w with respect to the basis {w1, w2} (In particular, your answer will show that w is an element of W)\r
\n" ); document.write( "\n" ); document.write( "w1=(2, -1, 4)
\n" ); document.write( "w2=(5,2,-3)
\n" ); document.write( "w=(-6, -15, 40) \n" ); document.write( "

Algebra.Com's Answer #505611 by Fombitz(32388) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"w=a%2Aw%5B1%5D%2Bb%2Aw%5B2%5D\"$
\n" ); document.write( " $\"w%5Bx%5D=a%2A2%2Bb%2A5=-6\"$
\n" ); document.write( " $\"w%5By%5D=a%2A%28-1%29%2Bb%2A2=-15\"$
\n" ); document.write( "1. $\"2a%2B5b=-6\"$
\n" ); document.write( "2. $\"-a%2B2b=-15\"$
\n" ); document.write( "Multiply eq.2 by 2 and add to eq. 1,
\n" ); document.write( " $\"2a%2B5b-2a%2B4b=-6-30\"$
\n" ); document.write( " $\"9b=-36\"$
\n" ); document.write( " $\"b=-4\"$
\n" ); document.write( "and
\n" ); document.write( " $\"-a%2B2%28-4%29=-15\"$
\n" ); document.write( " $\"-a-8=-15\"$
\n" ); document.write( " $\"a=7\"$
\n" ); document.write( "Now check to make sure that $\"w%5Bz%5D\"$ is consistent with this solution.
\n" ); document.write( " $\"w%5Bz%5D=a%2A4%2Bb%2A%28-3%29=40\"$
\n" ); document.write( " $\"%287%294%2B%28-4%29%28-3%29=40\"$
\n" ); document.write( " $\"28%2B12=40\"$
\n" ); document.write( " $\"40=40\"$
\n" ); document.write( "True.
\n" ); document.write( "So then,
\n" ); document.write( " $\"w=7w%5B1%5D-4w%5B2%5D\"$
\n" ); document.write( "So $\"w\"$ is a linear combination of the $\"w%5B1%5D\"$ and $\"w%5B2%5D\"$ and is therefore a member of $\"W\"$ . \n" ); document.write( "

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