Thread: Subspaces

I need to determine whether U is a subspace of V. If it is not a subspace, I need to state a condition that fails and give a counter example showing that that condition fails.

a) V is the space of all differentiable functions R->R , and U is the set of differentiable functions whose derivative at 0 takes the value 1.

b) V is the space of all polynomials with real coefficients, viewed as functions R->R, and U is the set of all differentiable functions R->R.

c) V=R^4, and U= {(a, ab, b, c) belonging to R^4, a,b,c belong to R}

I know to prove that these are subspaces of V I need to prove that it is closed under scalar multiplication and addition, but I'm not sure how to write them out.

Originally Posted by skittle

I need to determine whether U is a subspace of V. If it is not a subspace, I need to state a condition that fails and give a counter example showing that that condition fails.

a) V is the space of all differentiable functions R->R , and U is the set of differentiable functions whose derivative at 0 takes the value 1.

b) V is the space of all polynomials with real coefficients, viewed as functions R->R, and U is the set of all differentiable functions R->R.

c) V=R^4, and U= {(a, ab, b, c) belonging to R^4, a,b,c belong to R}

I know to prove that these are subspaces of V I need to prove that it is closed under scalar multiplication and addition, but I'm not sure how to write them out.

What have you tried?
for 1) let what is
and what does this tell you

for 2 just verify what you need for a subspace a polynomial can be written as


for 3 try adding two vectors of that form and see if there result is of the correct form.