Conditions for a system of linear forms to be a basis

[cat_56]

Hi, I need help with the following problem:

Let $V$ be the vector space of all polynomial functions from $\Bbb R [x] _{\le 3}$ to $\Bbb R [x] _{\le 3}$. Consider the linear forms $f_i$ defined for $p$ in $V$, as $f_i(p)=p(a_i)$, where $a_i \in \Bbb R$, for $i \in \{1,2,3,4\}$.


a) Determine under what conditions $\{f_1, f_2, f_3, f_4\}$ is a basis for $V^*$.


b) Suppose that you have found the conditions for a), determine the base $B$ of $V$ of which $\{f_1, f_2, f_3, f_4\}$ is the dual base.


Any suggestions? Thanks!

 

[Deleted member 4993]

Hi, I need help with the following problem:

Let $V$ be the vector space of all polynomial functions from $\Bbb R [x] _{\le 3}$ to $\Bbb R [x] _{\le 3}$. Consider the linear forms $f_i$ defined for $p$ in $V$, as $f_i(p)=p(a_i)$, where $a_i \in \Bbb R$, for $i \in \{1,2,3,4\}$.


a) Determine under what conditions $\{f_1, f_2, f_3, f_4\}$ is a basis for $V^*$.


b) Suppose that you have found the conditions for a), determine the base $B$ of $V$ of which $\{f_1, f_2, f_3, f_4\}$ is the dual base.


Any suggestions? Thanks!

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[stapel]

Let $V$ be the vector space of all polynomial functions from $\Bbb R [x] _{\le 3}$ to $\Bbb R [x] _{\le 3}$. Consider the linear forms $f_i$ defined for $p$ in $V$, as $f_i(p)=p(a_i)$, where $a_i \in \Bbb R$, for $i \in \{1,2,3,4\}$.

a) Determine under what conditions $\{f_1, f_2, f_3, f_4\}$ is a basis for $V^*$.

b) Suppose that you have found the conditions for a), determine the base $B$ of $V$ of which $\{f_1, f_2, f_3, f_4\}$ is the dual base.

Is this showing up as properly formatted on your computer? Because I'm seeing stuff with lots of dollar signs, underscores, and back-slashes...?