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Orthonormal vectors

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Three vectors v1, v2 and v3 are given (see attachment).

1. show that v1, v2 and v3 are orthonormal
2. vector w is given (see attachment). Find <w,v1> , <w,v2> and <w,v3>
3. Is w a linear combination of v1, v2 and v3? If yes, what is the combination? If not, why not?
4. is the given vector u (see attacment) a linear combination of v1, v2 and v3? Justify your answer.

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https://brainmass.com/math/matrices/matrix-theory-linear-combinations-orthonormality-209237

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The 6 pages solution discusees the concepts of orthonormality and inner product. It shows how to decompose a vector into a linear combination of basis vectors.

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A Two State System Spanned by Two Orthonormal Vectors

Consider a two state system spanned by two orthonormal vectors, |1> and |2>. The action of an operator  is defined via:

Â|1> = 2|1> + i|2>
Â|2> = -i|1> + 3|2>

Find Â|ѱ> where

|ѱ> = |1> + |2>

Can you now verify your answer for Â|ѱ> by doing the calculation in matrix representation?

And then compute the four "matrix elements":

A_ij ≡ <i|Â|j>

Where i, j = 1, 2.

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