ker(c^\ast \circ f) = \{v|v\in V,c^\ast (f(v))=0\} = \{v|v\in V,f(v)\in ker(c^\ast)\} the last identity comes from the definition of kernel.
As for the second part, I'm not sure if it can be applied to any space W, but if W is finite dimensional with dimension n, then every linear functional's...
If I understand correctly: f^T(c^\ast ) = c^\ast \circ f
so that: ker(f^T(c^\ast))=ker(c^\ast \circ f)
which is all the vectors of V that are mapped by f into the kernel of c*.
Or: f^{-1}(ker(c\ast)) in other words...
for the second part, you should show that f^{-1}(ker (c\ast)) is a linear...
write it out as a sum:
y=x'Ax = \sum_i \sum_j x_iA_{ij}x_j
\frac{\partial y}{\partial x_k} = \sum_i\sum_j(\left x_iA_{ij}\delta_{jk} + \delta_{ik}A_{ij}x_j \right)
if A is symmetric:
\frac{\partial y}{\partial x} = 2Ax
this works only if A is symmetric though, otherwise it would be...
A small correction:
For the matrix
H (H^TWH)^{-1}H^TW - I_{m \times m}
I want the absolute value of the diagonal elements to be larger than the absolute values of the off-diagonal elements.
That's because I want:
\forall(j\neq i) \delta\phi_i > \delta\phi_j \Rightarrow r_i > r_j
where...
I have the following problem:
I have a set of m measurements $\mathbf{\phi}$
and I estimate a set of 3 variables $\mathbf{x}$
The estimated value for $\mathbf{\phi}$ depends linearly on $\mathbf{x}$ : Hx=\Tilde{\phi}
The solution through weighted linear least squares is:
$\mathbf{x}$ =...
There is a theorem: http://en.wikipedia.org/wiki/Finitely_generated_abelian_group" [Broken]
which states that every finite Abelian group can be decomposed into the direct product of groups of prime order (any group of prime order is cyclic btw).
Let's look at the subgroup A_p. The order of...
I'm sorry, but if that's what generated means, the general element should be of the form:
\forall \ x\in \mathcal{S} , \ x=\displaystyle\sum_r (S_r \otimes v_r \otimes T_r
\otimes u_r \otimes U_r \ + \ S_r \otimes u_r \otimes T_r
\otimes v_r \otimes U_r)
where S_r,T_r,U_r \in \mathcal{F}(V)...
I currently self study from the book "A Course in Modern Mathematical Physics" by Peter Szekeres, and I'm currently reading the chapter on tensors, which he defines using the concept of Free Vector Spaces.
He gives a re-definition of Grassmann's algebras introduced in the previous section by...