Show that ds² = dx² + dy² + dz² - c²dt² is invariant under Lorentz transformation.
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- Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).Find a basis B for R3 such that the matrix for the linear transformation T:R3R3, T(x,y,z)=(2x2z,2y2z,3x3z), relative to B is diagonal.Let T:RnRm be the linear transformation defined by T(v)=Av, where A=[30100302]. Find the dimensions of Rn and Rm.
- Show that the image of the annulus {1<|z|<2} under w=2/(z-1) is the domain {W: Re(W)>-1, |W=2/3|>4/3} using Möbius transformation.Sketch the image of the rectangle with vertices at (0, 0), (1, 0), (1, 2), and (0, 2) under the specified transformation.T is the expansion represented by T(x, y) = (x, 6y).Use Laplace transformation to solve the IVP y"+y'=0, y(0)=1, y'(0)=1
- Sketch the image of the rectangle with vertices at (0, 0), (1, 0), (1, 2), and (0, 2) under the specified transformation.T is the expansion represented by T(x, y) = (2x, y).Let ℝ [x]≤ 2 ---> ℝ be a linear transformation such that T(x2)= -3, T(x2+x)=-4, T(x+1)=-1 What is T(ax2+bx+c) for orbitrary a,b,c ∊ℝ?Find the laplace transform of the equation and pls with complete solution and clear handwritting