I am given a set $T = \{u_1,\ u_2,\ ... \ ,u_k\}$, and I am another set $T' = \{w_1,\ w_2, \ ...\ ,w_k\} $, where $T'$ is obtained from applying the Gram-Schmidt process to orthogonalise and normalise the vectors in $T$. I am tasked to find the transition matrix $P$ from $T$ to $T'$.
I know what the Gram-Schmidt process does, and I have attempted to express each element of $T'$, $w_i$, as the expression as deduced from the corresponding elements of $T$.
But beyond that, I am completely stuck. I admit that I do not have a good grasp on the concepts of linear algebra at this level. Does anyone have any tips on how I can proceed? Thank you!
First, consider the fact that vectors in both $T$ and $T'$ are from the same subspace. Hence, we can express the vectors in $T$ as a linear combination of the vectors in $T'$. The general formula is given as follows:
$$u_k = (u_k \cdot w_1)w_1 + (u_k \cdot w_2)w_2 + \cdots + (u_k \cdot w_k)w_k$$
You can then express each of the vectors as a column matrix:
\begin{pmatrix} u_k \cdot w_1 \\ u_k \cdot w_2 \\ \vdots \\ u_k \cdot w_k \end{pmatrix}
The transition matrix can then be obtained by joining the column matrices together.