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Elementary Linear Algebra (MindTap Course List)

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.3: Subspaces Of Vector Spaces

Problem 49E

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complex variables, need help with showing all steps for conformal mapping to unit disk

2. Find a one-to-one conformal mapping f that maps each of the following regions onto the unit disk
D(0, 1).
(a) The sector {z € C: [argz| <

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Let T:RnRm be the linear transformation defined by T(v)=Av, where A=[30100302]. Find the dimensions of Rn and Rm.
Let T be a linear transformation T such that T(v)=kv for v in Rn. Find the standard matrix for T.
Let 1-R-R be a linear map defined by L (x.y.z) = (x+2y, 2x+ 3y+z,x+ y+3)· Then what is the kernel of ? OA KerL=((0.0.0).(-2,1.1)} OB. KerL= (a(-2.1.1) aER) OCKerL =((0.0.0)] OD. KerL=1(x.y.2) ER x+2y =0}: OE None of the choices in the list is correct.
4. Find the kernel and image of the linear map f(x,y, z) = (2x – z, y + 5z). Also, find the nullity and rank of the map f. Is f surjective? Is f injective? %3D
(3) Consider the linear map T : R² → R² which sends (1,0) → (-3,5) and (0, 1) → (4, – 1). (d) What is the pre-image of the upper half plane under T? Explain algebraically and draw a picture.
(3) Consider the linear map T : R² → R² which sends (1,0) → (-3,5) and (0, 1) → (4, – 1). (c) What is the image of the lower half plane under T? Explain algebraically and draw a picture.
In this problem we will graphically investigate properties of conformal maps. For each of the maps (x, y) → (u, v) below, complete parts i) through iii). (a)_u(x,y)= x² + y², v(x, y) = 2xy (b) _u(x, y) = x² – y², v(x, y) = 2xy -Y (c)_u(x, y) = e cosx, v(2,y) = e sinx i) Plot the curves u(x, y) = 0, 1 and v(x, y) = 0,1 in the x, y-plane. Do the curves intersect at right angles? ii) Based on your picture from part (a), is the map conformal? Why or why not? iii) Prove that the map f(x, y) = u(x, y) + iv(x, y) is/is not conformal using the Cauchy-Riemann conditions. If possible, represent the map by a function f(z) where z = x + iy.
Find the image of the circle |Z – 5| > 5 in the z – plane under the inverse transformation w = by representing both the regions graphically.
Let be a linear map defined by L(x,y.z) = (x+ 2y, 2x + 3y+ z,x+y+z)• Then what is the kernel of ? L:R³→R³ %3D O A. KerL = { (0,0,0), (-2.1,1)} %3D O B. KerL = {(0.0.0)}: OC. KerL= {(x.y,z) ERx+ 2y=0}: %3D %3D O D.None of the choices in the list is correct. O E. Kerl = {a (– 2,1,1) | a ER)
Find the image of the closed disk |z| ≤ 1 under the given linear mapping w = f(z) and represent the linear mapping with a sequence of plots such as translation, magnification, rotation. *f(z)=3iz
Let E and F be two vector spaces with the same finite dimension dim(E) = dim(F) For every linear map f: EF, the following properties are = n. equivalent: (a) f is bijective. (b) f is surjective. (c) f is injective.. (d) Ker f = (0).
3. Let V be a complex inner product space and T: V → V a linear transformation. Suppose that T satisfies the following property (T(u), u) = 0 Show that T = 0. Vu € V₂

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