(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. The attempt at a solution

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

And if M and N are finite dimension subspaces then [itex]dim(M+N)=dim M + dim N- dim(M \cap N)[/itex]. The diemnsion of the subspace generated by p1 and p2 is 2, and so is the dimension of the subspace generated by q1 and q2. Am I right? But then from this [itex]dim(M+N)=dim M + dim N- dim(M \cap N)[/itex] I get that [itex](M \cap N)[/itex] has a dimension of 1.

Thank you!

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Polynomial subspaces

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