6
votes
Accepted
Jacobian of the normalization of vector x, with respect to vector x.
$\def\vec{\operatorname{vec}}$
$\def\qty#1{\left( #1 \right)}$
$\def\d{{\sf d}}$
There is no need to write the Jacobian matrix as $\frac{\partial \vec {\bf y}}{\partial \vec {\bf x}}$ since both $\bf ...
- 1,001
5
votes
Accepted
A dual space to a space of homogeneous polynomials
The dual can be described in terms of differential operators. Given a polynomial $p(x, y)$ in two variables we can consider the family $f_{i, j}$ of linear functionals defined by taking the partial ...
- 433k
3
votes
Jacobian of the normalization of vector x, with respect to vector x.
Using product differential rule:
$$
d\frac{\textbf{x}}{\sqrt{\langle \textbf{x}, \textbf{x} \rangle}}=\frac{d\textbf{x}}{\sqrt{\langle \textbf{x}, \textbf{x} \rangle}}
+
\textbf{x}\:d\frac{1}{\sqrt{\...
- 986
3
votes
Accepted
Additive inverse in a Field $\Bbb Z_2$ with 2 elements 0 and 1
The notation $\mathbb{Z}_n$ refers to the integers modulo $n$. This is the remainder after dividing by n. For example, $19 \mod 4 = 3$ because $19 = 4\cdot 4 + 3.$ So, $\mathbb{Z}_2$ looks at ...
- 9,798
1
vote
Accepted
Low-dimension matrix approximation?
$E_2W$ is a matrix with rank at most $n_2$. Following the Eckart-Young-Mirsky theorem, the smallest possible result for the objective function is obtained when $E_2 W$ is equal to the truncated SVD of ...
- 228k
1
vote
How axioms of inner product ensure that an instantiation/realization capture notion of angle correctly?
A real inner product on a real vector space $V$ is a symmetric positive definite bilinear form $V \times V \to \mathbb{R}$.
You already accept that $|x| = \sqrt{\langle x, x\rangle}$ captures the ...
- 9,355
1
vote
Ideal , subring and subspace of a linear algebra over a field
The definition of a linear algebra in Hoffman-Kunze is what is more commonly known as an (associative) algebra over a field. It seems that you are also assuming the existence of identity, which would ...
- 3,347
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