document.write( "Question 356729: What is a subspace ? How do you prove that it is a subspace ?\r
\n" ); document.write( "\n" ); document.write( "I know that it is a straight line or plane that passes through the origin.
\n" ); document.write( "But the proof of a subspace of 3 rules seems too basic.
\n" ); document.write( "It almost allows all vectors to be subspaces. I have not seen a vector that is not a subspace yet.\r
\n" ); document.write( "\n" ); document.write( "the rules are something like multiply by 0
\n" ); document.write( " addition of u and v scalars
\n" ); document.write( " multiplication by scalars.\r
\n" ); document.write( "\n" ); document.write( "My assignment question reads
\n" ); document.write( "W is the space of all vectors of the form (x, y, x-y) Find out if W is a subspace. JUstify your answer.\r
\n" ); document.write( "\n" ); document.write( "I need the answer put in basic english, with as much detail as possible. The question somehow is worth 10% of assignment, and to me it seems to basic. I just have problem with the wording.
\n" ); document.write( "Can anyone help with expressing this answer in clear English.\r
\n" ); document.write( "\n" ); document.write( "Thank You \n" ); document.write( "

Algebra.Com's Answer #254659 by robertb(5830) $\"\"$ $\"About$

You can put this solution on YOUR website!
Assuming addition of vectors and scalar multiplication in $\"R%5En\"$ , two conditions had to be met for a subset to be a subspace:
\n" ); document.write( "i) If u and v are vectors in W, then u + v is in W, and
\n" ); document.write( "ii) If u is in W, and c is a a scalar, then c*u is in W.
\n" ); document.write( "For the question above,
\n" ); document.write( "i) (x, y, x-y)+(z, w, z-w) = (x+z, y+w, x-y+z-w) = (x+z, y+w, (x+z)-(y+w)),
\n" ); document.write( "ii)c*(x, y, x-y) = (cx, cy, c(x-y)) = (cx, cy, cx-cy).
\n" ); document.write( "Therefore W is a subspace of $\"R%5E3\"$ . \n" ); document.write( "

\n" );