V=M(sub n)(R), and S is the subset of all n x n matrices with det(A)=0.
V=M(sub n)(R), and S is the subset of all skew-symmetric matrices.
V=R^5, and S is the set of vectors (x1, x2, x3) in V satisfying x1-7x2+x3=6
V is the vector space of all real-valued functions defined on the interval [a,b], and S is the subset of V consisting of those functions satisfying f(a)=f(b).
V=P5, and S is the subset of P5 consisting of those polynomials satisfying p(1)>p(0).
V=R^2, and S is the set of all vectors (x1, x2) in V satisfying 6x1+7x2=0.
V=P3, and S is the subset of P3 consisting of all polynomials of the form p(x)=ax^3+bx
V=M(sub n)(R), and S is the subset of all skew-symmetric matrices.
V=R^5, and S is the set of vectors (x1, x2, x3) in V satisfying x1-7x2+x3=6
V is the vector space of all real-valued functions defined on the interval [a,b], and S is the subset of V consisting of those functions satisfying f(a)=f(b).
V=P5, and S is the subset of P5 consisting of those polynomials satisfying p(1)>p(0).
V=R^2, and S is the set of all vectors (x1, x2) in V satisfying 6x1+7x2=0.
V=P3, and S is the subset of P3 consisting of all polynomials of the form p(x)=ax^3+bx
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No, not closed under addition
Yes
No, not closed and doesn't contain the zero
Yes
No, doesn't contain the zero
Yes
Yes, but you have to allow a and/or b to be any real including zero
Yes
No, not closed and doesn't contain the zero
Yes
No, doesn't contain the zero
Yes
Yes, but you have to allow a and/or b to be any real including zero