Question

Expanding a vector in a new basis.

There is a solved problem in my textbook which I don't really get. I have a row vector that has to be expanded in a new basis. So, I multiply each of the components of the vector with the basis vectors but for some reason multiplying those vectors produces a sum, instead of a matrix. Why is that?

Answer 1

Here is the problem:

Answer 2

\[|V\rangle = \left(\begin{matrix}1+i \\ \sqrt(3)+i\end{matrix}\right)\]

to be expanded into
\[|I\rangle =\frac{ 1 }{ \sqrt(2) }\left(\begin{matrix}1 \\ 1\end{matrix}\right)\]
\[|II\rangle =\frac{ 1 }{ \sqrt(2) }\left(\begin{matrix}1 \\ -1\end{matrix}\right)\]

Answer 3

So, \[|V\rangle = v_I |I\rangle+v _{II} |II\rangle\]
and
\[v_I=\langle I|V\rangle = \frac{ 1 }{ \sqrt(2) }(1\; \;1 )\left(\begin{matrix}1+i \\ \sqrt(3)+i\end{matrix}\right)\]

so on the next step they say that the answer is
\[v_I=\frac{ 1 }{ \sqrt(2) }(1+\sqrt(3)+2i)\]

Answer 4

Maybe a picture helps.

Answer 5

so we could add up a+b to get there in one coordinate basis or we could add up u+v to get there from the other coordinate basis.