what is the significance of poles and zero and wht is vector space?
534979
In complex analysis, a pole of a holomorphic function is a certain type of simple singularity that behaves like the singularity 1/z^n at z = 0. A pole of the function f(z) is a point z = a such that f(z) approaches infinity as z approaches a.\n\nZero or nothing is an important mathematical concept denoting absence which has facilitated the development of Maths and Science as we know them.\n\nA vector space is a set of vectors of the same dimensions bound by the laws and properties common to all of them
733734
sa·li·ent  (sl-nt, slynt)\nadj.\n1. Projecting or jutting beyond a line or surface; protruding.\n2. Strikingly conspicuous; prominent. \n\nSalient Pole: A motor has salient poles when its stator or field poles are concentrated in to confined arcs and the winding is wrapped around them (as opposed to distributing them in series of slots).
50645
if we speak of numbers and not of limits, the answer is no without exceptions: 0 is defined in group theory as the null element, so it is BY DEFINITION the element which multiplied by any other gives itself: always 0.\nInstead, what we usually assume as always true, but is not true in some finite algebraic models, is that if A*B=0 then either A=0 or B=0. This is not always true in all algebraic models (for exemple in modular algebra).
115853
#include <iostream>\n#include <vector>\nusing namespace std;\n#define rep(i, n) for((i) = 0; (i) < (n); (i)++)\n#define rep2(i, k, n) for((i) = k; (i) < (n); (i)++)\nvoid f21();\nint n;\ndouble mat[100][100];\nint main()\n{\nwhile(true)\n{\ncin >> n;\nif(!n) break;\nmemset(mat, 0, sizeof(mat));\nf21();\n}\nreturn 0;\n}\n\nvoid f21()\n{\ncout << "****************************************" << endl;\ncout << "*          INVERSE OF MATRIX           *" << endl;\ncout << "****************************************" << endl;\n\ncout << "ORDER OF THE MATRIX: " << n << endl;\ncout << "ENTER ELEMENT ROW-WISE" << endl;\nint m = 2*n;\nint i, j, k;\nrep(i, n) {\nrep(j, n) {\ncin >> mat[i][j];\ncout <<"&#92;t";\nprintf("%5.3lf", mat[i][j]);\n}\nrep2(j, n, m) {\nif(i == (j-n)){\nmat[i][j] = 1;\ncontinue;\n}\nmat[i][j] = 0;\n}\ncout << endl;\n}\n\ncout << endl << "INVERSE MATRIX: " << endl;\ndouble pivot, comm;\nrep(i, n) {\npivot = mat[i][i];\nif(mat[i][i] == 0){\ncout << "DIVISION BY ZERO." <<endl;\nreturn;\n}\nrep(j, m) mat[i][j] = mat[i][j]/pivot;\nrep(j, n) {\nif(i == j) continue;\ncomm = mat[j][i];\nrep2(k, i, m) {\nmat[j][k] = mat[j][k] - (comm*mat[i][k]);\n}\n}\n}\n\nrep(i, n) {\nrep2(j, n, m) {\ncout <<"&#92;t";\nprintf("%6.4lf", mat[i][j]);\n}\ncout << endl;\n}\ncout << endl;\n}\n\n/*This is writen in C++ this will output the inverse of a Matrix\ninput:\n4\n1 2 3 4\n2 1 4 2\n2 5 7 12\n1 -1 3 11\n\n5\n1.173 -0.634 0 0 0\n-0.539 1.173 -0.634 0 0\n0 -0.539 1.173 -0.634 0\n0 0 -0.539 1.173 -0.634\n0 0 0 -0.539 1.173\n\n0\n*/
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The vector potential is defined only to within the gradient of a scalar. That is, you can add the gradient of any scalar field to the answer below and get an equally valid expression for the vector potential.\n\nIf B is the constant (vector) value of the magnetic field then the vector potential is:\n\nA(r) = -r x (B/2)\n\nwhere r is the the position vector and x means cross product. In Cartesian coordinates in which the magnetic field is pointing in the direction of the z axis, the three components of A are:\n\nA_x = -(B/2)y\nA_y = (B/2)x\nA_z = 0\n\nwhere B here is a constant number.
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in this case, the cross-product uses the sine of the angle between the two vectors to multiply the product of the magnitudes of the two vectors.   the angle between any vector and itself is zero.\n\nthe sine of zero is zero.\n\nthen, i x i = |i|*|i|*sin(0) = 1*1*0 = 0
100903
If A is an n-by-n symmetric positive definite matrix with real entries, that means ((v transpose)Av) > 0 for nonzero real n-by-1 column vectors v.\n\nConsider the 3x3 case with A equal to\n\n[ a b c ]\n[ b d e ]\n[ c e f ]\n\nLet u, v, and w be the nonzero column vectors\n\n[ 1 ]\n[ 0 ]\n[ 0 ]\n\n[ 0 ]\n[ 1 ]\n[ 0 ]\n\n[ 0 ]\n[ 0 ]\n[ 1 ]\n\nrespectively.  Then\n\n((u transpose)Au) = a > 0\n((v transpose)Av) = d > 0\n((w transpose)Aw) = f > 0\n\nCan you see how to generalize this for other values of n besides three and/or for matrices and vectors with complex entries?
9229
Given a vector space V, over a field F, and a linear function A:V -> V (that is for x,y in V and 'a' in F then  A(ax+y)=aA(x)+A(y) ), e in F is an eigenvalue of A if there is a non-zero vector v in V such that A(v)=ev. \n\nThe more specific meaning used in a linear algebra class is, if you have a NxN matrix A then e (where e is a rational/real/complex number) is an eigenvalue for A if there is an N dimensional vector v such that Av=ev.  \n\nEigenvalue comes from the German meaning proper value.
489205
0 is the cardinal of the empty set. Thats the mathematic rigurous definition. Before that, zero was defined conceptualy, as "no object", "nothing", "none", in order to define the quantity/measure of "nothing".
1161312
To the best of my knowledge.The point is a well defined mark. It can be located anywhere in space and its position can be specified excatly wrt some co ordinate system. So it is not undefined.\nHowever the other 2 elements have one thing in common and that is we cannot specify the dimensions of any of them. In other words each of them can be extended to infinite dimensions.
95774
The RANK of matrix A, an m x n matrix, is the dimension of the row spaces or the column space. Or, the number of basic variables in Ax = 0. \n\nUsually written as rank A.\n\nAlso, for m x n matrix A, the row space and the column space have the same dimension, \n\nwritten dim R(A) = dim C(A).
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In the following: x, y, z real vectors and c real scalar\n\n\n- scalar product: x.y = x1*y1 + x2*y2 + x3*y3\n\n- euclidian distance: √x.y\n\n\nThere are 4 properties deriving from scalar product:\n\n\nsymmetry\n\n1: x.y == y.x\n\n\nlinearity\n\n2: (x+z).y == x.y + z.y\n\n3: (c*x).y == c(x.y) == x.(c*y)\n\n\npositive definiteness\n\n4: for all x: x.x ≥ 0 and  x.x == 0 iff x == 0\n\n\n\nEuclidian distance is a specific definition of a vector norm || . ||:\n\nEuclidian norm: ||x|| =  √x.x\n\n\nThe following holds for each scalar product induced norm: \n\nSchwarz's inequality\n\n5: |x.y| ≤ ||x||*||y||\n\n\nTriangle inequality\n\n6: | ( ||x|| - ||y|| ) | ≤ ||x + y|| ≤ ||x|| + ||y||\n\n\n\nRemark: | . | acts on real numbers and || . || on vectors
1008877
In cartography, as in all mappings, a vector implies direction & length (or force). I'm not familiar with other uses of the word... I usually have used the term in connection "cross-impact matricies." Hope that was helpful.
664853
One basic idea for analyzing a mathematical object is to look at morphisms into 'simple' objects of that type.\n\nFor example, a group G operating on a set S is a homomorphism of G into the symmetric group Sym(S). We regard the symmetric groups as 'simple' objects that we can use to understand other groups.\n\nA group representation is a homomorphism of the group G into the group gl(V) of invertible linear transformations on some vector space V. The good thing about this is that you get all of linear algebra to help you understand the group gl(V) and its subgroups, and hence to understand the group G. The homomorphism is called a representation of G.\n\nAmong the questions that can be asked:\n1. Are there any invariant subspaces of V? (an invariant subspace is one that is preserved by every linear transformation in the image of G).\n2. Can the vector space V be written as a direct sum of invariant subspaces in some nice way?\n3. Are there enough representations to distinguish all elements of the group?\n4. What happens if you look at representations in vector spaces over different base fields?\n\nIf your group is a Lie group (a differentiable manifold that is also a group), the you can look at smooth homomorphisms. It turns out the there is a vast amount of geometry that appears. This geometry is closely related to problems in particle physics.\n\nThis is just the beginning of the study of representations. There is much,much more. Yes, it is fun!
303805
I don't see how components are necessary.  \n\n"r" and "v" are just shorthand for r(t) and v(t), assuming we are looking at position (r) and velocity (v) as time varying "vectors".  And v(t) is just the time derivative of r(t) (by DEFINITION), that is, v(t) is DEFINED as r'(t).\n\nSo there isn't any actual CONTENT in expanding them into components. We are just choosing different ways to symbolize the quantities involved and there isn't anything to prove.
260979
Assume the vectors (x,y,z) and (u,v,w) are elements of this set and check that they are closed under the vector space operations (that under those operations, the resultant elements also satisfy the equation).\nAs a matter of fact, as a subset of R^3, it is not a subspace as it does not contain the 0 vector. However, it is a 2-dimensional vector space if you give it its own coordinate system. It is properly an affine subspace of R^3.
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1.www.physics.uoguelph.ca/tutorials/vectors/vectors.html \n2.mathforum.org/~klotz/Vectors/vectors.html