0
votes
1answer
29 views

Linear Algebra,Conjugate Transpose

Let $ M_n(\mathbb C) $ be the space of all $ n\times n $ matrices with complex entries. Prove that function $ \langle, \rangle : M_n(\mathbb C) \times M_n(\mathbb C) \to \mathbb C $ defined by $ ...
3
votes
1answer
63 views

Geometric intuition behind subspaces in $\mathbb C^n$

While learning elementary linear algebra one develops a great deal of geometric intuition in $\mathbb R^n$. It helps to see the forest for the trees and leads through proofs. After meeting ...
0
votes
1answer
28 views

Inner product over the $C^2$

Let a, b, c, d ∈ C and consider the vector space $C^2$ Suppose inner product is defined as: $⟨x, y⟩ = ax_1\bar y_1 + bx_2\bar y_1 + cx_1\bar y_2 + dx_2\bar y_2$ I am trying to find all a, b, ...
0
votes
2answers
38 views

Cauchy-Schwarz in complex case, using discriminant

There is a proof of the real case of Cauchy-Schwarz inequality that expands $\|\lambda v - w\|^2 \geq 0 $, gets a quadratic in $\lambda$, and takes the discriminant to get the Cauchy-Schwarz ...
0
votes
2answers
38 views

Proving an inequality with the Schwarz inequality

Given a vector space with a Hermitian dot product defined, prove the following inequality using the Schwarz inequality. Let $f$ be a complex value function that is continuous within $0 \le x \le 1$, ...
1
vote
3answers
57 views

Inverse of complex vector

How is the inverse of a complex vector calculated? In $\Bbb R$, the inverse vector $ X= \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \\ ...
1
vote
1answer
285 views

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 ...
1
vote
1answer
52 views

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 ...
0
votes
0answers
43 views

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 $M$ ...
2
votes
0answers
87 views

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 ...
2
votes
1answer
96 views

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 ...
5
votes
2answers
444 views

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 ...
0
votes
1answer
34 views

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 - ...
4
votes
3answers
301 views

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 ...
3
votes
1answer
175 views

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 ...
10
votes
3answers
287 views

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 ...
2
votes
1answer
38 views

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 ...
2
votes
2answers
104 views

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 ...
7
votes
1answer
323 views

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 ...
4
votes
2answers
321 views

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 ...
6
votes
3answers
667 views

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: ...
2
votes
2answers
2k views

What are the rules for complex-component vectors and why?

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 ...
2
votes
3answers
1k views

Geometric interpretation of the multiplication of complex numbers?

I've always been taught that one way to look at complex numbers is as a cartesian space, where the "real" part is the x component and the "imaginary" part is the y component. In this sense, these ...