Question

find the vector x determined by the given coordinate vector [x]B and the given basis B

B=[3, -5][4, 6] XB=[5,3]

Answer 1

let's call basis of B is b1=[3.-5] and b2= [4.6]
we need the coordinate vector [x]_B, that means x = m b1 + n b2, find m, n
now, just replace all we know
\[x=\left[\begin{matrix}5\\3\end{matrix}\right]\]
\[b_1=\left[\begin{matrix}3\\-5\end{matrix}\right]\]
\[b_2=\left[\begin{matrix}4\\6\end{matrix}\right]\]
what we don't know, let them there and find them out (that's the goal of the work)
so that we have
\[x = mb_1+nb_2\]
\[\left[\begin{matrix}5\\3\end{matrix}\right]=m\left[\begin{matrix}3\\-5\end{matrix}\right]+n\left[\begin{matrix}4\\6\end{matrix}\right] \]
\[\left[\begin{matrix}5\\3\end{matrix}\right]=\left[\begin{matrix}3m\\-5m\end{matrix}\right]+\left[\begin{matrix}4n\\6n\end{matrix}\right] \]
so that, we can take line by line like
5 = 3m + 4n
3= -5m +6n
solve for m, n , you will have m = 9/19 and n = 17/19
conclusion: \[[x]_B=\dfrac{9}{19}\left[\begin{matrix}3\\-5\end{matrix}\right]+\dfrac{17}{19}\left[\begin{matrix}4\\6\end{matrix}\right]\]

Answer 2

I don't force you to study. You are free!! I don't give you the stuff for free. I review myself. Although it violates the rule of the site, I give it out to help you and help myself also. If you want to study, take it as an example. Find out what you don't understand and ask me about that. If not, it's perfectly ok to me. :)

Answer 3

I just having a hard time understanding this, thanks for helping though