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I cannot determine the approach to solve problems like this, so any help will be appreciated. I need help specifically with the third term on the right, as in the equation to be solved.

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I would be grateful if someone can prove this in a detailed manner.

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  • $\begingroup$ In the future, please take the time to enter important parts of your question as text instead of pasting a picture. Images are neither searchable nor accessible to people using screen readers, nor do they appear in summaries. $\endgroup$ – amd Apr 26 '18 at 0:28
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I think you should go and review the Gram-Schmidt method.

What the problem says is: You want a basis for the space spanned by $ U, x, v $. You can add to the basis by first computing the component of $ v $ orthogonal to $ U $, call it $ w $,then the component of $ x $ orthogonal to $ U $, let's call it $ z $, and then the component of $ z $ orthogonal to the vector $ w $.

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