Abstract Vector Space Question

[epkid08]

Homework Statement

Let $g_1(t) = t - 1$ and $g_2(t)= t^2+t$. Using the inner product on $P_2$ defined in example 10(b) with $t_1=-1,t_2=0,t_3=1$, find a basis for the orthogonal complement of $Span(g_1, g_2)$.

Homework Equations

From example 10(b)
$\langle p, q \rangle = \sum_{i=1}^{k+1} p(t_i)q(t_i)$

$(p,q)\in P_2$

The Attempt at a Solution

Well the orthogonal complement of $span(g_1,g_2)$ will be $x$ such that $x\cdot (c_1g_1 + c_2g_2) = 0$, but how can I find the basis for that set? And why and where do we need to use inner product? I am confused

 

[Mark44]

Homework Statement

Let $g_1(t) = t - 1$ and $g_2(t)= t^2+t$. Using the inner product on $P_2$ defined in example 10(b) with $t_1=-1,t_2=0,t_3=1$, find a basis for the orthogonal complement of $Span(g_1, g_2)$.

Homework Equations

From example 10(b)
$\langle p, q \rangle = \sum_{i=1}^{k+1} p(t_i)q(t_i)$

$(p,q)\in P_2$

The Attempt at a Solution

Well the orthogonal complement of $span(g_1,g_2)$ will be $x$ such that $x\cdot (c_1g_1 + c_2g_2) = 0$, but how can I find the basis for that set?

Start with this equation. Here, x $\in$ P₂, so you can write it as a + bt + ct².

Where you show the dot product in the equation above, you should be using the inner product from example 10b. Also, put in the formulas for the functions g₁ and g₂.

And why and where do we need to use inner product? I am confused