Questions tagged [inner-products]
For questions about inner products and inner product spaces, including questions about the dot product.
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Timelike vector equivalence relation
I'm a little bit stuck with the demonstration of the equivalence relation between timelike vectors, specifically in the transitive property. This is what i have:
$g(v,w)<0, g(w,z)<0,g(v,v)<0,...
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why isn't cos used in this case with dot product? (coordinates) [closed]
example
why isn't cos used in this equation for coordinates. i am sorry if this question is a bit dumb, but i really don't understand why here cos is used, but above it's not.
3
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Show that function is positive definite
In the process of showing that the function\begin{equation}
\langle \mathbf{z}, \mathbf{w}\rangle=\overline{z_1}w_1+(1+i)\overline{z_1}w_2+(1-i)\overline{z_2}w_1+3\overline{z_2}w_2
\end{equation} is ...
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Assumption for maximum rank
Consider two sets of vectors $\left\{x^i\right\}_{i=1}^N$ and $\left\{y^j\right\}_{j=1}^n$, where $\left\{x^i\right\}_{i=1}^N$ are linearly independent vectors, $x^i \in \mathbb{R}^d$ and $y^j \in \...
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Can one solve this complex linear equation
Suppose the equations are as follows:
$$\text{tr}\{AH\}= c$$
where $c \in \mathbb{C}$ and $A$ is a symmetric matrix in $M_{N}(\mathbb{R})$ are both known and $H$ is a Hermitian matrix which looks like:...
- 11
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Interpretation of $\max \sup$
I am reading the regret analysis proof of LinUCB given in Lattimore's Bandit Algorithms. He makes the following assumption:
$$ \max\limits_{t\in[n]}\sup\limits_{a,b\in\mathcal{A}_t} \langle\theta^* , ...
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Orthonormal basis for $\mathbb{C}^2$ over $\mathbb{R}$ [closed]
$\mathbb{C}^2$ is a 4-dimensional vector space over $\mathbb{R}$ with basis $\left\{\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} i \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix}, ...
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Why do we have 2 types of products for vectors?
Why is it that we have two types of products for vectors ? Why do we have one scalar product and one vector product? If we were concerned with the magnitude of the product, wouldn't it be easier to ...
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Jacobian and Gradient map multiplication vs dot product confusion in model
Hi I have a very simple model and I'm trying to learn the math of it.
Basically, I have an input matrix X (m x n). An output matrix Y (m x n) is formed from some convolution H. The figure of merit is ...
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Graphical Intuition of a Linear Transformation in terms of Row Vectors
The graphical intuition of a linear transformation (matrix) $A \in \mathbb{R}^{m \times n}$ applied on a vector $\textbf{v}$ in terms of the column vectors $\textbf{c}_i$ of $A$ is quite clear to me:
...
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Show that $ \left( \int_1^\infty f \right)^2 \leq \int_1^\infty x^2(f(x))^2 dx $ [duplicate]
In Axler's Linear Algebra Done Right (4e), Chapter 6A (Inner Product Spaces), exercise 18 is as follows:
(a) Suppose $f: [1, \infty) \to [0, \infty)$ is continuous. Show that
$$
\left( \int_1^\infty ...
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A new identity for the scalar triple product in $\mathbb{R}^3$
Using the determinant identity for the scalar triple product for $[(\vec{a} \times \vec{b}).\vec{c}]^2$ and factorizing using the quadruple scalar product identity $(\vec{p} \times \vec{q}).(\vec{r} \...
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How to prove an Identity regarding the Norm of the Second Derivative of a Curve
Let $\gamma:A\rightarrow\mathbb{R}^{3}$ be a normal differentiable curve.
Suppose $\gamma(t)=\left(\begin{matrix}x(t)\\y(t)\\z(t)\end{matrix}\right)\in{C}^{1}(A)\;$. Suppose also that $\forall{t}\in{A}...
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Why is the inner product space defined separately?
While learning about the inner product space, I became curious
why it is defined separately?
In my opinion, there seems to be no difference between defining the inner product space separately and ...
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Trouble finding norm of $T: H_1 \to H_2$, which is defined by $T(x)=\sum_{i=1}^n\lambda_i \langle x, a_i \rangle b_i \quad \text{for each } x\in H_1$.
Let $H_1$ and $H_2$ be complex Hilbert spaces. Let $\lambda_1, \lambda_2, \ldots, \lambda_n$ be complex numbers, and let $\{a_1, a_2, \ldots, a_n\} \subset H_1$ and $\{b_1, b_2, \ldots, b_n\} \subset ...
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