# transition matrix of dual basis

Q :Let $$\left\lbrace v_i \right\rbrace^n_{i=1}$$ and $$\left\lbrace w_i \right\rbrace^n_{i=1}$$ be basis of V and also let $$\left\lbrace \phi_i \right\rbrace^n_{i=1}$$ and $$\left\lbrace \sigma_i \right\rbrace^n_{i=1}$$ be basis of $$V^{\ast}$$ and dual basis to $$\left\lbrace v_i \right\rbrace^n_{i=1}$$ and $$\left\lbrace w_i \right\rbrace^n_{i=1}$$ respectively. Suppose, P is the change of basis matrix from $$\left\lbrace v_i \right\rbrace^n_{i=1}$$ to $$\left\lbrace w_i \right\rbrace^n_{i=1}$$ , then show that $$(P^{-1})^T\text{ is the change of basis matrix from }\left\lbrace \phi_i \right\rbrace^n_{i=1} \text{to } \left\lbrace \sigma_i \right\rbrace^n_{i=1}$$

My approach :By hypothesis $$\exists P,\text{ a transition matrix from }\left\lbrace v_i \right\rbrace^n_{i=1}$$ to $$\left\lbrace w_i \right\rbrace^n_{i=1}$$. That is $$w_1=a_{11}v_1+a_{12}v_2+\cdots+a_{1n}v_n$$
$$w_2=a_{21}v_1+a_{22}v_2+\cdots+a_{2n}v_n$$
$$\vdots$$
$$w_n=a_{n1}v_1+a_{n2}v_2+\cdots+a_{nn}v_n$$ $$P=\begin{pmatrix}a_{11} & a_{21}&\cdots&a_{n1}\\\ \vdots & \vdots&&\vdots\\\ a_{n1}& a_{2n}&\cdots& a_{nn} \end{pmatrix}\\$$ Again similar way, $$\exists Q,\text{ a transition matrix from }\left\lbrace \phi_i \right\rbrace^n_{i=1}$$ to $$\left\lbrace \sigma_i \right\rbrace^n_{i=1}$$. That is
$$\sigma_1=b_{11}\phi_1+b_{12}\phi_2+\cdots+b_{1n}\phi_n$$
$$\sigma_2=b_{21}\phi_1+b_{22}\phi_2+\cdots+b_{2n}\phi_n$$
$$\vdots$$
$$\sigma_n=b_{n1}\phi_1+b_{n2}\phi_2+\cdots+b_{nn}\phi_n$$ $$Q=\begin{pmatrix}b_{11} & b_{21}&\cdots&b_{n1}\\\ \vdots & \vdots&&\vdots\\\ b_{n1}& b_{2n}&\cdots& b_{nn} \end{pmatrix}\\$$Now by definition of dual basis we know,$$\sigma_i(w_j)=\delta_{ij} = \begin{cases} 1, i=j \\ 0, i\neq j \end{cases}$$. Now \begin{align*} \sigma_i(w_j)&=(b_{i1}\phi_1+\cdots+b_{in}\phi_n)(a_{j1}v_1+\cdots+a_{jn}v_n)\\ &=b_{i1}\phi_1(a_{j1}v_1+\cdots+a_{jn}v_n)+\cdots+b_{in}\phi_n(a_{j1}v_1+\cdots+a_{jn}v_n) \\ &=b_{i1}a_{j1}+\cdots+b_{in}a_{jn} \end{align*} But here I stuck. I am not found anything to link up with P to Q. Besides, please give me some mathematical notation which help me to reduce my solution size if any. Any hint or solution will be appreciated .

I'll use your notation for the basis vectors, but for the coefficients, I'll write them as $$P_{ij}$$ and $$Q_{ij}$$ rather than $$a_{ij}$$ and $$b_{ij}$$. The key observation which allows you to link the two matrices is that for any $$f \in V^*$$, we can expand it as $$$$f = \sum_{i=1}^n f(v_i) \cdot \phi_i$$$$ (i.e the $$b_{ij}$$'s which you wrote can actually be written in terms of $$\sigma_j$$). To see why this is true, temporarily call the RHS $$T$$. Then, for every $$j$$, (by definition of dual basis) it is immediate that $$T(v_j) = f(v_j)$$. So we have shown that $$f$$ and $$T$$ agree on the basis $$\{v_1, \dots, v_n\}$$; hence they are equal everywhere. Another result we need is that $$P^{-1}$$ is the transition matrix from $$\{w_1, \dots , w_n\}$$ to $$\{v_1, \dots, v_n\}$$. Now, we have a few lines of computation, and the only thing to be careful of is the indices.

For every $$1 \leq j \leq n$$, we have that \begin{align} \sigma_j &= \sum_{i=1}^n \sigma_j(v_i) \cdot \phi_i \\ &= \sum_{i=1}^n \sigma_j \left(\sum_{k=1}^n \left( P^{-1}\right)_{ik}\, w_k\right) \cdot \phi_i \\ &= \sum_{i=1}^n \left( P^{-1}\right)_{ij} \cdot \phi_i \end{align} The last equality is because the $$\sigma$$'s are dual to the $$w$$'s. On the other hand, since we said $$Q$$ is the transition matrix from the $$\phi$$'s to $$\sigma$$'s, this means, for every $$1 \leq j \leq n$$, we have $$$$\sigma_j = \sum_{i=1}^n Q_{ji} \cdot \phi_i.$$$$

Notice how the indices of $$i$$ and $$j$$ in the two summations are reversed. Thus, it follows that $$Q = \left( P^{-1}\right)^t$$.

(You may be interested to know that this question is in fact a special case of a much more general theorem regarding the matrix representation of a linear map with respect to given bases and the matrix representation of the dual map with respect to the corresponding dual bases. If you want to learn more about this, I highly recommend reading Linear Algebra, by Friedberg, Insel and Spence; in particular Section 2.6 on dual spaces, and Theorem 2.25 in that section.)

• Thanks for a nice explanation and I will definitely look your suggested book. Thanks again @peek-a-boo – emonhossain May 22 at 12:05