Linear Algebra II - Orthogonal Basis If $X = [1,-2, 1, 6]$ in $\mathbb{R}^4$ and let $U = span \{[2,1,3,-4], [1,2,0, 1]\}$.
 How would you show that $\{[1,0,2,-3], [4,7,1,2]\}$ is another orthogonal basis of $U$.
Would I be able to simply go ahead by constructing if the second set of vectors is orthogonal by using $\{E1,E2\}$? Or would i have to incorporate $U$ somehow?
 A: You have to show $2$ things:
1) $B = \{[1,0,2,-3], [4,7,1,2]\}$ is a basis for $U$
2) B is an orthogonal set of vectors
Obviously, 2) is the easiest part here since $1*4 + 0*7 + 2*1 + (-3)*2 = 0$, implying that these $2$ vectors are orthogonal.
For 1), one has to prove that:


*

*These two vectors span $U$

*These two vectors are linear independent 


Linear independency should be straightforward to prove (make a linear combination of the two vectors that is equal to $[0,0,0,0]$ and show the scalars must be $0$).
That $U$ is spanned by these 2 vectors might be more difficult, but the easiest way to show this is that you can write $[2,1,3,-4],[1,2,0,1]$ as linear combination of the two given vectors, because then you know $span\{[1,0,2,-3], [4,7,1,2]\} = span\{[2,1,3,-4],[1,2,0,1]\} = U$
A: Hint 
Stuff the vectors spanning the spaces into two matrices $$A = \left[\begin{array}{cccc}2&1&3&-4\\1&2&0&1\end{array}\right], B = \left[\begin{array}{cccc}1&0&2&-3\\4&7&1&2\end{array}\right]$$
To show that a basis is orthogonal: show that all basis vectors are pairwise orthogonal to each other. How can you do that with matrix multiplication?
To show they span the same space, search for a matrix $T$ so that $TB = A$
If you manage to solve and find such a $T$ then it is the matrix transforming coordinates form the first basis into the other.
