Let T = {u1,u2, ...,uk} be a basis for a subspace V in Rn and T′ = {w1,w2,...,wk} be an orthonormal basis of V obtained from T by applying the Gram-Schmidt process with normalisation so that each wi is a unit vector. Find the transition matrix P from T to T′. Give your answer in terms of u1,u2, ...,uk and w1,w2, ....,wk.
How should I attempt this question?
first you need to define $w_1,w_2,...,w_n$ using Gram-Schmidt process
let $v_1=w_1$ and $w_2=v_2-c_{12}v_1$ where $c_{12}=\frac{<v_2,w_1>}{<w_1,w_1>}$ it is obvious that
$<w_1,w_2>=0$ now we can continue to our process
$w_3=v_3-c_{13}w_1-c_{23}w_2$
$w_4=v_4-c_{14}w_1-c_{24}w_2-c_{34}w_3$
$\vdots$
$w_n=v_n-c_{1n}w_1+...+c_{n-1n}w_{n-1}$
where $c_{ij}=\frac{<v_j,w_i>}{<w_i,w_i>}$
$\begin{pmatrix} 1 & 0 & 0 & 0 & \cdots & 0\\ -c_{12} & 1 & 0& 0 & \cdots & 0 \\ -c_{13}+c_{12}c_{23} &-c_{23} & 1 & 0 & \cdots & 0\\ -c_{14}+c_{12}c_{24}+c_{13}c_{34}-c_{12}c_{23}c_{34} & -c_{24}+c_{23}c_{34} & -c_{34} & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots& \vdots & \vdots & \\ -c_{1n}+c_{12}c_{2n}+...+c_{1n-1}c_{n-1n}-c_{12}c_{23}c_{3n}-...+(-1)^{n+1}c_{12}c_{23}...c_{n-1n} & \cdots & \dots & \cdots & -c_{n-1n}&1 \end{pmatrix}$ $\begin{pmatrix} v_1 \\ v_2 \\ v_3 \\ v_4\\ \vdots \\ v_n \end{pmatrix}$ $=$ $\begin{pmatrix} w_1 \\ w_2 \\ w_3 \\ w_4 \\ \vdots \\ w_n \end{pmatrix}$