How would I do this problem? (these are all column vectors)
Find the coordinate vector $[\mathbf{v}]_b$ is $\mathbf{v}=\left[3, 5, -4\right]$ and $B$ is the orthonormal basis $$B=\{[\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}}],[-\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}}],[[0,-1,0]\}$$
I tried to take the first component of v and multiplied it by the first component of B, and did the same for the others, but that did not seem to work.
Best Answer
You should find the triple $(l_1,l_2,l_3)$ s.t $v=l_1 [\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}}]+l_2 [-\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}}]+l_3 [[0,-1,0]$. In other words, you have to solve the following 3x3 system.
$\frac{1}{\sqrt{2}} l_1-\frac{1}{\sqrt{2}}l_2+0\cdot l_3=3$
$0\cdot l_1+0\cdot l_2-1\cdot l_3=5$
$\frac{1}{\sqrt{2}}l_1+\frac{1}{\sqrt{2}}l_2+0\cdot l_3=-4$
So the requested vector is $[l_1,l_2,l_3]$.