(Solved): (5 pts.) Consider the subset S={(x,y,z)inR^(3)|z^(2)=x^(2)+y^(2)} of R^(3). Is S a vector subspace o ...
(5 pts.) Consider the subset
S={(x,y,z)inR^(3)|z^(2)=x^(2)+y^(2)}of
R^(3). Is
Sa vector subspace of
R^(3)? (
20pts.) Let
C(R)denote the vector space of all continuous real valued functions on
Rand for
k>=1let
C^(k)(R)denote the vector spaces of all
k-times continuously differentiable real valued functions on
R. Determine if the given subsets
Sof
C(R)and
C^(k)(R)are subspaces or not. (a)
S={finC(R)|f(7)=0}(b)
S={finC(R)|f(7)=3}(c)
S={finC(R)|f(7)f(3)=0}(d)
S={finC^(1)(R)|f^(')(2)=0}(e)
S={finC^(1)(R)|f^(')(2)=5}(f)
S={finC^(1)(R)|f^(')(2)f(5)=0}(g) |
=0for all
{:xinR}(h) for all
xinR(a) (5 pts.) Let
P_(3)(R)denote the vector space of polynomials with real coefficients of degree
n<=3and consider the subset
S={a_(0)+a_(1)x+a_(2)x^(2)+a_(3)x^(3)inP_(3)(R)|a_(0)+a_(1)+a_(2)+a_(3)=1}That is, consider the subset
Sof all polynomials of degree
n<=3whose coefficients sum to 1 . Note that
Sis NOT a subspace
P_(3)(R)since the zero polynomial 0 is not in
S. Show that
Sis NOT closed under addition or scalar multiplication in general. (b) ( 5 pts.) Next show that if we change the condition on the coefficients of polynomials defining the subset
Sof polynomials to
S={a_(0)+a_(1)x+a_(2)x^(2)+a_(3)x^(3)inP_(3)(R)|a_(0)+a_(1)+a_(2)+a_(3)=0}then
Sis a subspace of
P_(3)(R). 4. (a) (5 pts.) Let
M_(2\times 2)(R)denote the vector space of all
2\times 2matrices with real number entries and consider the subset
S={AinM_(2\times 2)(R)|det(A)=1}That is, consider the subset
Sof all
2\times 2matrices with real number entries and determinant equal to 1 . Note that
Sis NOT a subspace
M_(2\times 2)(R)since the zero matrix 0 is not in
S. Show that
Sis NOT closed under addition or scalar multiplication algebraically and use your work to find counterexamples to the closure properties. (b) (5 pts.) What happens if we change the the condition on the determinants defining the subset
Sto
S={AinM_(2\times 2)(R)|det(A)=0}?