Question #45761

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.

{

(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.

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