# Tagged Questions

An inner product space is a vector space equipped with an inner product. The inner product is a generalisation of the "dot" product often used in vector calculus.

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### How do different inner products give different angles?

I know that for each inner product $\langle , \rangle_{A}$ on $\mathbb{R}^n$, there is an associated positive definite symmetric matrix $A$ so that $\langle x,y \rangle = x^{T}Ay$. I was wondering if ...
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### Find angle between 2 vectors (inner products)

For complex vector spaces, i.e. vector spaces with scalars from the field $C$of complex numbers, inner products must have slightly different properties. To see why, consider the following vectors ...
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### Orthogonality in Hilbert Spaces

For the sake of concreteness, let's say that our Hilbert space is the beloved $\mathscr L^2(\Bbb R)$. Suppose that we have $\psi,\phi\in\mathscr L^2(\Bbb R)$, what's the intuitive meaning to a ...
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### Is it possible to compute the volume of a cone on a inner product space?

This is a matter of curiosity for me. Volumes are often compute using triple integration. But is it possible to compute volumes on a vector space with an inner product defined on that vector space?
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### The probability of measuring the control qubit in zero in a quantum circuit

I’m working on an assignment where I have to solve some questions about a quantum circuit. In particular, I have a quantum circuit with three qubits: $|0\rangle$(referenced to as the control qubit), ...
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### What does “canonical” mean in vector space?

I was watching this video: https://www.youtube.com/watch?v=RDkwklFGMfo And the professor is talking about the inner product... then he brings up the "canonical" representation of the inner product in ...
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### Orthogonal projection by standard norm of and a non-standard norm in $\mathbb{R}^2$

Suppose a closed convex set $S\subset \mathbb{R}^2$ is given by by the convex hull of $(0,1) (-1,1), (-1,0), (1,0)$ and a continuous, convex and decreasing curve $F$ linking $(1,0)$ and $(0,1)$, ...
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### Is there only one way to define a norm from an inner product?

Given an inner product $\langle,\rangle$, we can define a norm by $||x|| = \langle x,x \rangle^{\frac{1}{2}}$. My question is, are there other ways to derive a norm from an inner product space and if ...
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### P is an orthogonal projector if and only if $P^2 = P$ and $AP$ is symmetric

I'm given that the inner product in a linear subspace $V \subset R^n$ is defined as $<X,Y>_V = X'AY$ where matrix $A$ is positive definite. I need to show that $P$ is an orthogonal projector if ...
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### maximum vector in $\Bbb R^n$ which any two of them has dot product less than 0

I'm thinking of in $\Bbb R^n$, at most how many vectors can we have any two of them's dot product are less than 0 In $\Bbb R^3$, I guess that there are at most $4$ vector but I can't prove it.(at ...
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### Why is the inner product of a quadratic form a quadratic form?

I was going through a derivation of the second derivative of the $\log \det X$ where $X$ is symmetric positive definite, I noticed that despite the second order approximation of log det is written as: ...
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### How can you use the Law of Cosines to show that “u ∙ v = ║u║║v║cosθ ” [duplicate]

How can I do that? I didn't know there was a relationship between those 2?
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### Completeness of metric, normed and inner product spaces

For a metric space to be complete, it needs to have all cauchy sequences converge in the metric. 1) For a normed space to be complete, does it need Cauchy sequences to converge in the norm or in the ...