1. ## Dimension and basis

Let $\displaystyle M$ be a subspace of the vector space $\displaystyle \mathbb{R}_2[t]$ generated by $\displaystyle p_1(T)=t^2+t+1$ and $\displaystyle p_2(T)=1-t^2$, and $\displaystyle N$ be a subspace generated by $\displaystyle q_1(T)=t^2+2t+3$ and $\displaystyle q_2(T)=t^2-t+1$. Show the dimension of the following subspaces:$\displaystyle M+N$, $\displaystyle M \cap N$, and give a basis for each.

2. ## Re: Dimension and basis

Originally Posted by noricka
Let $\displaystyle M$ be a subspace of the vector space $\displaystyle \mathbb{R}_2[t]$ generated by $\displaystyle p_1(T)=t^2+t+1$ and $\displaystyle p_2(T)=1-t^2$, and $\displaystyle N$ be a subspace generated by $\displaystyle q_1(T)=t^2+2t+3$ and $\displaystyle q_2(T)=t^2-t+1$. Show the dimension of the following subspaces:$\displaystyle M+N$, $\displaystyle M \cap N$, and give a basis for each.

3. ## Re: Dimension and basis

Originally Posted by Drexel28
I have tried the following: if I take the linear combination of $\displaystyle p_1$ $\displaystyle p_2$ $\displaystyle q_1$ $\displaystyle q_2$, I get $\displaystyle (a+b+c+d)t^2 + (a+2c-d)t +(a+b+3c+d).$ And a basis of this polynomial is $\displaystyle \{1,t,t^2\}$, which means the dimension of M+N is 3.

And if M and N are finite dimension subspaces then $\displaystyle dim(M+N)=dim M + dim N- dim(M \cap N)$. But what is the dimension of M and N?

4. ## Re: Dimension and basis

how many basis elements do M and N have?

5. ## Re: Dimension and basis

Both of them have 3?

6. ## Re: Dimension and basis

how can a set generated by 2 elements have 3 basis vectors???

to be fair, p1 and p2 are only listed as generating elements, its not explicitly stated whether or not {p1,p2} forms a basis.

but, by the very definition of "generate" they are elements of a span-set. are they linearly independent? (if you decide they are, how can you be sure?

and what does this mean dim(M) is?)

7. ## Re: Dimension and basis

p1 and p2 are linearly independent, so their dimension is 2?

8. ## Re: Dimension and basis

elements do not have dimension, vector spaces (and their subspaces) have dimension.

the dimension of a vector space, V, is defined to be the number of elements in ANY basis.

if S is a subset of V, then dim(span(S)) ≤ |S|.

if S is a linearly independent set, then dim(span(S)) = |S|.

9. ## Re: Dimension and basis

So the dimension of the subspace M generated by p1, and p2 is 2, then. And that's the same with N as well.

10. ## Re: Dimension and basis

Was my idea of the dimension of M+N right? And what about the bases?