Find the dual basis 4

Question

Let $P_3$ be the real vector space consisting of all polynomials with real coefficients of degree less than $3$ the set $\{5, x-1, x^2-1\}$ is a basis of $P_3$.

Find the dual basis of the above basis

Answer 1

Recall that if $V$ is a vector space, $V^*=\text{Hom}_{\mathbb{R}}(V,\mathbb{R})$. Moreover, if $v_1, \dots , v_n\in V$ is a basis of $V$, the dual basis $\left\{v_1^*, \dots , v_n^*\right\}$ is defined by $$v_i^*(v_j)=\delta_{i,j}.$$ Indeed for any linear map it suffices to know it on a basis to know it everywhere.

So here you have for example that $(x-1)^*(5)=0$, $(x-1)^*(x-1)=1$ and $(x-1)^*(x^2-1)=0.$ Thus $(x-1)^*:P_3\rightarrow \mathbb{R}$ defined as above.

Can you determine the matrix of the linear map $(x-1)^*$ w.r.t. the given basis in $P_3$ and the standard basis of $\mathbb{R}$? Do the same for the other two maps.