# A problem on vector space( included in the attached image)

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.

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

• 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. – amd Apr 26 '18 at 0:28

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 $w$.