What's the relation between complex and vectors......Complex are usually represented as position vectors and during rotation we only rotate the directed line formed by the two complex.Why do they share some properties when they are entirely different?
49063
What is the relation  between complex numbers and vectors?  I have read in some places "a complex number a 2-dimensional vector".  One can easily observe that $i\cdot i=-1$ in complex multiplication like the cross product of vectors.    Is  vector only the generalisation of complex number to n dimensions?   Is there any fundamental difference between those two?
31190
I haven't taken any complex analysis course yet, but now I have this question that relates to it.  Let's have a look at a very simple example. Suppose $x,y$ and $z$ are the Cartesian coordinates and we have a function $z=f(x,y)=\cos(x)+\sin(y)$. However, now I change the $\mathbb{R}^2$ plane $x,y$ to complex plane and make a new function, $z=\cos(t)+i\sin(t)$.  So, can anyone tell me some famous and fundamental differences between complex plane and $\mathbb{R}^2$ by this example, like some features $\mathbb{R}^2 $ has but complex plane doesn't or the other way around. (Actually I am trying to understand why electrical engineers always want to put signal into the complex numbers rather than $\mathbb{R}^2$, if a signal is affected by 2 components)   Thanks for help me out!
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Now I'm confused with what "a linear transformation" means.  In linear algebra textbook, I learned that a linear transformation is $T:V \to W$, where V,W are vector spaces, which satisfies additivity and homogeneity, in other words, $T(u+v)=Tu+Tv, T(av)=aTv$ for all $u,v \in V$ and $a \in F$  But in my complex analysis textbook, $\displaystyle f(z)=\frac{az+b}{cz+d}$, $a,b,c,d \in \mathbb C$, is introduced as an example of a linear transformation.   However, this function $f$ doesn't seem to follow the definition from linear algebra. Indeed, $f(0) \neq 0$.  Is it like there are two kinds of linear transformations in mathematics, or they are actually the same thing but I don't get it well?
79500
The fact that the dot product and the cosine of the angle between two vectors are mutually computable is easy to show (see the two sides in the two answers at ).  But looking at the dot product, I would never have thought that it somehow captures something about the angle (and vice versa).  How did the connection get discovered? Who were the major players? Did it just fall out of the development of matrix operations for linear algebra (or did the dot product come first) or are these only related by hindsight or what?
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Possible Duplicate:          On Wikipedia, it says that:     Matrix representation of complex numbers   Complex numbers $z=a+ib$ can also be represented by $2\times2$ matrices that have the following form: $$\pmatrix{a&amp;-b\\b&amp;a}$$   I don't understand why they can be represented by these matrices or where these matrices come from.
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My professor keeps mentioning the word "isomorphic" in class, but has yet to define it... I've asked him and his response is that something that is isomorphic to something else means that they have the same vector structure. I'm not sure what that means, so I was hoping anyone could explain its meaning to me, using knowledge from elementary linear algebra only. He started discussing it in the current section of our textbook: General Vector Spaces.  I've also heard that this is an abstract algebra term, so I'm not sure if isomorphic means the same thing in both subjects, but I know absolutely no abstract algebra, so in your definition if you keep either keep abstract algebra out completely, or use very basic abstract algebra knowledge, that would be appreciated.
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I have just been exposed to rotation matrices, and it is showing extremely strong connection with group theory and differential geometry which I am both totally inept at.  Can someone in simple term explain what it means for a group to have a manifold structure and how this affects the mathematics of rotation group?
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If $a+ib$, $c+id$, $e+if$ are three complex numbers, than can we tell which one is greater or smaller between them? If yes, then how and if no then why not?   Can somebody give explanation on this.... I will be grateful to him.
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I would like to ask a question about affine matrix transformations, specifically a mathematical explanation about why these two ways to interpret them are equivalent.  Suppose I have a translation matrix T (for simplicity 2D) and a rotation matrix R as follows:  $$  T= \begin{bmatrix}         1 &amp; 0 &amp; tx \\         0 &amp; 1 &amp; ty \\         0 &amp; 0 &amp; 1 \\         \end{bmatrix}  R= \begin{bmatrix}         \cos \theta &amp; -\sin \theta &amp; 0 \\         \sin \theta &amp; \cos \theta &amp; 0 \\         0 &amp; 0 &amp; 1 \\         \end{bmatrix} $$  Suppose: $$ tx = 0 , ty = 2, \theta = 45^{ \circ } $$  Now if I transform a quad applying translation and rotation to every vertex in this order: R * T * Vertex (Column vector), I obtain this:      Since both translation and rotation transforms are done around the origin, considering T first and R after. However this is equivalent to considering R first and T after modifying the quad relative to itself and not the origin:       The result is the same. Why is this equivalent?
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I was just wondering how complex numbers can be applied in electrical engineering and why we use complex numbers over regular, real numbers for this application (e.g what capabilities does the complex number have that real numbers do not in electrical engineering)?  I have done some research concerning impedances and understand how they are written in complex form, however I am still confused why complex numbers are necessary in this field over regular numbers.
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I understand how to manipulate vector spaces and subspaces and how to prove various statements about them, but I still don't fully understand what they represent geometrically.  I just need an intuitive grasp as to what these are. Is a vector space a geometric area that contains all possible vectors of the field $\Bbb F$? For example, if $V$ is a vector space over $\Bbb R^3$ then does that mean $V$ contains all vectors in three-dimensions that are part of $\Bbb R$? But then what is a subspace of $V$? Would that perhaps be a plane? Would it be a vector space in $\Bbb R^2$?
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As normed division algebras, when we go from the complex numbers to the quaternions, we lose commutativity. Moving on to the octonions, we lose associativity. Is there some analogous property that we lose moving from the reals to the complex numbers?
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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 complex numbers are like vectors, and they can be added geometrically like normal vectors can.  However, is there a geometric interpretation for the multiplication of two complex numbers?  I tried out two test ones, $3+i$ and $-2+3i$, which multiply to $-9+7i$.  But no geometrical significance seems to be found.  Is there a geometric significance for the multiplication of complex numbers?
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I apologize if this question is dumb, but I've looked all over for a straightforward answer, and either I can't find one or either the terms are too complex for me to understand. I have only a rudimentary knowledge of Mechanics, but I do understand basic Linear Algebra. So torque, mathematically, is the cross product of the radial distance vector and a force vector. This cross-product gives another vector that is orthogonal to both vectors and it points either outside or towards the &quot;page&quot; (in the context of a two-dimensional diagram). Assuming this is correct, I do not understand what it pointing in or out means. Does it even have a physical, intuitive meaning? The best answer I've been able to come up with is that it's just a mathematical convention with no actual physical meaning, meant to provide a framework within which operations between torque vectors, such as addition and subtraction, make sense. Am I correct or way off the mark here?
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I got the difference between polar vectors and axial vectors (pseudovectors).   An example of pseudovector is $\mathbf{B}$. But why exactly the magnetic field is a pseudovector and its components parallel to an coordinate axis do no change signs if I reverse that axis?
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Im trying to get my head around complex integration/complex line integrals. Real integration can be thought of as the area under a curve or the opposite of differentiation. Thinking of it geometrically as the area under a curve or the volume under a surface in 3 dimension is very intuitive.   So is there a geometric way of thinking about complex integration? Or should I just be viewing it as process that reverses differentiation? Or has integration other meanings in complex analysis? Here is an example, could someone explain this to me - Here's the definition of the integral along a curve gamma in C, parameterized by $w:[a, b] -&gt;C$ \begin{equation} \int_\gamma f(z)dz = \int_a^b f(w(t)).w{(t)}'dt \end{equation}   So I have -  $\gamma$ is the unit circle with anti-clockwise orientation parameterized by  $w:[0, 2\pi]\to C$  $w(t) = e ^{it} = Cos(t) + iSin(t)$  So if use the definition of the integral,   $\int_\gamma f(z)dz = \int^b_a f(w(t)).w{(t)}&#39;dt$,  and work this out it comes to  $\int^{2\pi}_0 i dt = 2{\pi}i$  So what does this $2{\pi}i$ represent? Does it mean anything geometrically, like if a regular integral works out to be 10 that means the area under the curve between 2 points is 10...
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Hi my course specifically talks about :  Cartesian and Polar Coordinates in 3 Dim, second Degree eqns in 3 vars, reduction to canonical forms, straight lines, shortest distance between 2 skew lines, Plane, sphere, cone, cylinder, paraboloid, ellipsoid,hyperboloid of one and two sheets and their properties.  This is merely indicative, and I would really like if the book talks about these (atleast) and some more. Would appreciate if it contained solved examples and exercise problems as well  Help Much Appreciated, Soham