Calculus: transformation of complex numbers

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• Let a ε complex numbers and distinct from 0.

  Prove that the eigenvectors of the transformation T : Complex numbers^2 -> complex numbers^2 given by T ( z , w ) = ( z + aw , w) span on 1-dimensional subspace of complex numbers^2 and give a basis for this subspace.

• Consider the linear transformation T : complex numbers^n -> complex numbers^n given by T( z1, z2, ... , zn ) = ( a1z1, a2z2, ... , anzn). What is the dimension of the subspace spanned by the eigenvectors of T? Exhibit a basis for this space, and give the eigenvalues.

  217190 Consider the Linear Transformation T: Complex Numbers Consider the linear transformation T : complex numbers^n -> complex numbers^n given by T( z1, z2, ... , zn ) = ( a1z1, a2z2, ... , anzn).

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  not be defined at z=-1, ie., z+1 or a multiple of it should be in the denominator So, let us suppose that where a and b are any two complex numbers We use This implies So, solving these two equations, we get So, hence So we define

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  To take this coincidence into account we can now use a new parametrization of the Mobius transformation: let ab = AB = m and c-a = C-A = n, where m and n are some complex numbers.

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  This problem demonstrates the properties of complex number and the transformation between Cartesian and polar representations.

• Properties of complex numbers

  263621 Properties of complex numbers Use De-Moivre's theorem to show that cos(5x) = 16cos^5(x) - 20cos^3(x)+5cos(x) sin(5x) = 5cos^4(x)sin(x)-10cos^2(x)sin^3(x)+sin^5(x) given the complex numbers z1 = (3+4i)/(3-4i) and z2=[(1+2i)/(1-3i)]

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  215505 Transformation Matrix in Vector Calculus Section 6.1 The Geometry of Maps from R^2 to R^2. Please see the attached pdf. file. Let D* be the parallelogram with vertices...Find a T such that D is the image set of D* under T.

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  576640 Linear Transformations, Matrices, Orthogonal Projections R stands for the field of real numbers. C stands for the field of complex numbers. 1.