Answer to Question #45761 in 3D CAD Modeling for sapna

Question

a) Let V =
{
(a; b; c; d ) 2 R
4
ja + b + 2c + 2d = 0
}
and
W =
{
(a; b; c; d ) 2 R
4
ja = b; c = d
}
. Check that V and W are vector spaces.
Further, check that W is a subspace of V . (4)
b) Find the dimensions of V and W . (3)
c) Let P
3
=
{
ax
3
+ bx
2
+ cx + d ja; b; c; d 2 R
}
. Check whether f (x) = x
2
+ 2x + 1 is in
[S], the subspace of P
3
generated by S =
{
3x
2
+ 1; 2x
2
+ x + 1
}
. If f (x) is in [S], write
f as a linear combination of elements in S. If f (x) is not in [S], find another
polynomial g(x) of degree at most two such that f (x) is in the span of S [ fg(x)g. Also
write f as a linear combination of elements in S [ fg(x)g.