# Subspaces

• February 6th 2011, 03:48 PM
skittle
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.
• February 6th 2011, 03:57 PM
TheEmptySet
Quote:

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 $f,g \in U$ what is
$f'(0)+g'(0)$ and what does this tell you

for 2 just verify what you need for a subspace a polynomial can be written as
$\displaystyle p(x)=\sum_{k=0}^{n}a_kx^k$

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