# Tagged Questions

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### Complex Numbers and Transformations

If a transformation t acts by rotating every point of the plane around the origin by $\pi/5$ clockwise and then proceeds to translate it by vector $v$ = $(1,2)$. How do I describe this ...
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### Complex/real base vectors

If I have a subspace of $\mathbb C^n$ which is spanned by $N$ complex basis vectors. Could I span the same space with $N$ basis vectors that each have real components? (but, of course, using complex ...
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### Bounded complex sequences and vector spaces

Let $\mathcal{l}_\infty$ be the vector space of all complex sequences that are bounded, i.e. if a sequence $(a_n)$ is in $\mathcal{l}_\infty$ if and only if there exists a non-negative real number $M$ ...
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### Set of all odd complex polynomials - complex vector space

Is the set of all odd complex polynomials a complex vector space? I'm given the following definition of a vector space: A vector space $V$ over the field $\mathbb F$ is a set $V$ of vectors, a field ...
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### Two quick eigenvalues & complex numbers questions

A) For a vector $v\in\mathbb{C^n}$, is $Im(-v)=Im(\overline{v})$ ? ($Im(v)$denoting the imaginary part of the vector $v$) My understanding: since every row of the vector is a complex number (say ...
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### Does anyone know any resources for Quaternions for truly understanding them?

I've been studying Quaternions for a week, on my own. I've learned various facts about them but I still don't understand them. My goal is to understand rotation quaternions specifically. I don't want ...
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### Need help interpreting an equation from an article (related to quaternions).

At this link, about half way down the page, there is an equation I don't understand http://physicsforgames.blogspot.com/2010/02/quaternions-why.html This is the equation. VV† = -x^2I^2 - y^2J^2 - ...
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### How do you construct the quaternion and the multiplication rules, like Hamilton did?

So, I understand complex number multiplication, and how it represents $2D$ rotations. What I don't understand is, how you add two more imaginary numbers $j$ and $k$, and get $3D$ rotations. I believe ...
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### The multiplication of 2D vectors produces what?

I am trying to learn about rotation quaternions, and in the process I am currently looking at 2D vector multiplication. To avoid confusion with other types of multiplication, this is the basic form I ...
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### square root of $1/2 + \sqrt3/2?$

Playing with Maple, I noticed that it gives the square root of $c = 1+\frac{\sqrt3}{2}$ as equal to $a = \frac{1}{2}+\frac{\sqrt3}{2}$. Indeed it checks out. But I got curious: how can I find that ...
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### Bound on unit vectors

could someone help me with this simple problem. As always with homework, hints are specially welcome. Let $v=(v_1,v_2)$ be a two-dimensional unit vector with complex coefficients. If $|v_1|<a$ and ...
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### Multiplying Complex Numbers by i

But I am wondering why isit $PQ \perp QR$ and not $QP \perp QR$ as shown below? UPDATE How do I get the equation: $(i-1)b=ic-a=i(1-2i)-(-1+4i)=3-3i$? Where does $(i-1)$ come from? I dont ...
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### Is there a “good” way to visualize complex vectors?

We often represent complex numbers as vectors in $\mathbb{R}^2$ with $x$ being the real axis and $y$ being the imaginary axis. We often represent 2-dimensional vectors over $\mathbb{R}$ in a similar ...
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### The real part treated like an angle in complex vector spaces

In my current lecture I regularly encounter usage of the real part of, say, a scalar product of two vectors similar to angles in classical geometry. For example in Hilbert space theory: Let $H$ be a ...
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### What's the Difference Between a Vector and an Hypercomplex Number?

What's the difference between a vector and an hypercomplex number? For instance a 4-vector and a quaternion. They seem to share many properties. Perhaps this question could be put more generally as: ...
I want to take the inverse of a dot product, where both vectors have complex components. In other words, if $\textbf{A} \cdot \textbf{B} = d$, and I know $\textbf{A}$ and $d$, I want to find a ...