Practice question · Put in order
Order the steps of the Gram-Schmidt process applied to a basis v1, v2, v3.
- Subtract from v3 its components along both w1 and w2 to obtain w3
- Subtract from v2 its component along w1 to obtain w2
- Divide each w by its own norm to obtain an orthonormal set
- Take w1 = v1 as the first direction
Hints
- Each new vector must be cleaned against every direction already fixed.
- Normalising is done once at the end, after all the directions are orthogonal.
Show the answer
- Take w1 = v1 as the first direction
- Subtract from v2 its component along w1 to obtain w2
- Subtract from v3 its components along both w1 and w2 to obtain w3
- Divide each w by its own norm to obtain an orthonormal set
Why
Each step removes exactly the part of the new vector that the previous directions already reach, so what remains is orthogonal to all of them. Normalising last keeps the arithmetic clean; doing it early forces you to carry square roots through every subtraction.
Practise Inner Product Spaces and Orthogonality
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