Find the adjoint Choose one from he following list of inner products and then find the adjoint of:
$$ \left[
  \begin{array}{ c c }
     1 & 2 \\
     -1 & 3
  \end{array} \right]
$$
When your inner prod cut is used on both its domain and target space.
a. The Euclidean dot product
b. The weighted inner product $\langle v,w\rangle=2v_1w_1+3v_1w_1$
I guess I don't really understand how to find the domain or target space of a matrix.
All I could find online was that the adjoint is the transpose of the cofactor matrix but I'm not sure how to calculate the cofactor matrix.
Any help is greatly appreciated!
 A: If you have an inner product $<\cdot, \cdot>$, the adjoint of an operator $A$ is $A^*$ where $<Ax,y> = < x,A^*y>$ for any $x,y$. 
For the dot product, $(Ax) \cdot y = (A x)^T y = x^T A^T y = x^T (A^T y) = x \cdot (A^T y)$ so the adjoint of $A$ is $A^T$. 
Now, try the other inner product on your own.
A: The adjoint with respect to the standard dot product is the transpose.
For the other dot product, first find its Gram matrix $B$:
$$
B=\begin{bmatrix}
\langle e_1,e_1\rangle & \langle e_1,e_2\rangle \\
\langle e_2,e_1\rangle & \langle e_2,e_2\rangle
\end{bmatrix}
=\begin{bmatrix}
2 & 0 \\
0 & 3
\end{bmatrix}
$$
(I suppose the formula is $\langle v,w\rangle=2v_1w_1+3v_2w_2$).
The dot product can then be realized as
$$
\langle v,w\rangle=v^TBw
$$
The adjoint to $A$ is the unique matrix $A^*$ such that, for all $v,w$, we have
$$
\langle v,A^*w\rangle=\langle Av,w\rangle
$$
that is, in matrix terms
$$
v^TBA^*w=(Av)^TBw
$$
or, in other terms
$$
(v^TBA^*-v^TA^TB)w=0
$$
Since this must hold for every $w$, we have
$$
0=v^T(BA^*-A^TB)
$$
and finally, since this holds for all $v$,
$$
BA^*=A^TB
$$
which means
$$
A^*=B^{-1}A^TB
$$
Note that the formula holds also for the standard product, where the Gram matrix is the identity.
A: This matrix acts on a $2$-dimensional vector (so it has a $2$-dimensional domain) and the output is a $2$-dimensional vector (so it has a $2$-dimensional target space (also called codomain)). So, if you're working in real numbers $\Bbb{R}$ (which I imagine is the case here), the domain is $\mathbb{R}^2$ and the target space is $\Bbb{R}^2$.
Whenever you want to find the domain and target space of a matrix, reason as above.
