Orthonormalize the set of the vectors

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Solution

Your input: orthonormalize the set of the vectors v1=⎡⎣⎢⎢⎢⎢⎢⎢⎢23−2313⎤⎦⎥⎥⎥⎥⎥⎥⎥
, v2=⎡⎣⎢⎢⎢⎢⎢⎢⎢2313−23⎤⎦⎥⎥⎥⎥⎥⎥⎥ using the Gram-Schmidt process.
According to the Gram-Schmidt process, uk=vk−∑j=1k−1projuj(vk)
, where proju(v)=u⋅vu⋅uuThe normalized vector is ek=ukuk⋅uk−−−−−−√
⋅
is the dot product operator.

Step 1

u1=v1=⎡⎣⎢⎢⎢⎢⎢⎢⎢23−2313⎤⎦⎥⎥⎥⎥⎥⎥⎥
e1=u1u1⋅u1−−−−−−√=⎡⎣⎢⎢⎢⎢⎢⎢⎢23−2313⎤⎦⎥⎥⎥⎥⎥⎥⎥

Step 2

u2=v2−u1⋅v2u1⋅u1u1=⎡⎣⎢⎢⎢⎢⎢⎢⎢2313−23⎤⎦⎥⎥⎥⎥⎥⎥⎥

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e2=u2u2⋅u2−−−−−−√=⎡⎣⎢⎢⎢⎢⎢⎢⎢2313−23⎤⎦⎥⎥⎥⎥⎥⎥⎥
Answer: the set of orthonormal vectors is e1=⎡⎣⎢⎢⎢⎢⎢⎢⎢23−2313⎤⎦⎥⎥⎥⎥⎥⎥⎥
, e2=⎡⎣⎢⎢⎢⎢⎢⎢⎢2313−23⎤⎦⎥⎥⎥⎥⎥⎥⎥