Question about matrices (Gram Schmidt algorithm, QR factorization)

http://en.wikipedia.org/wiki/QR_decomposition#Example_3

We then rearrange the equations above so that the ai are on the left, using the fact that the ei are unit vectors:

How did they do this rearrangement?

They rearranged the equations to get, for example, from

u_3 = a_3 - \text{proj}_{e_1} a_3 - \text{proj}_{e_2} a_3

a_3 = u_3 + \text{proj}_{e_1} a_3 + \text{proj}_{e_2} a_3

and then used the definition of

u_i

and

\text{proj}_e a

in each case.

Edexcel A-Level Economics Paper 1 - Unofficial Mark Scheme - 18 May 2023

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