Find the vector $x $ determined by the given coordinate vector $[x]B$ and the given basis$ B$.
$B = {[2, 3],[0, 1]}$ , $[x]B = [-3, -6]$
I cant seem to understand this concept. A detailed explanation step by step would be great, thanks!
[Math] Find the vector x determined by the given coordinate vector [x]B and the given basis B.
linear algebramatricesvectors
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Best Answer
If $e_1,e_2$ is the canonical basis then your new basis is $$b_1=2e_1+3e_2$$ $$b_2=e_2$$
Then solving for $e_1,e_2$ we have $$e_1=\frac{1}{2}b_1-\frac{3}{2}b_2$$ $$e_2=b_2$$
So, if you want the vector $-3e_1-6e_2$ be expressed in the new basis, by subbing $e_1,e_2$, we get $$-3\left(\frac{1}{2}b_1-\frac{3}{2}b_2\right)-6b_2$$ which is $$-\frac{3}{2}b_1-\frac{3}{2}b_2.$$