**Problem 1 **Show that

*S *= {(*x,y*) ∈R^{2 }: *x*^{2 }*> y*}

is not a subspace of R^{2 }by finding vectors *u,v *∈ *S *such that *u *+ *v /*∈ *S*.

**Problem 2 **Let *V *be the vector space consisting of all differentiable functions *f *: (0*,*1) → (0*,*1). Show that

by checking the three criteria

The constant function zero is in *S*.

If *f,g *∈ *S*, then (*f *+ *g*) ∈ *S *also.

If *f *∈ *S *and *c *is a constant, then *cf *∈ *S *also.

**Problem 3 **Show that

*S *= {*A *∈ *M*_{2×2 }: *A*^{2 }is the zero matrix}

is not a subspace of *M*_{2×2}.

**Problem 4 **Find three 3 × 2 matrices which span the subspace

null(*A*)}

**Problem 5 **Let

*S *= {*f *∈ *P*_{3 }: *f*(1) = *f*(2)}

Determine the general form of a polynomial in *S*. Find three cubic polynomials which span *S*.