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