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- 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.Define T:R2R2 by T(v)=projuv Where u is a fixed vector in R2. Show that the eigenvalues of A the standard matrix of T are 0 and 1.Find the standard matrix for the reflection T of R3 in the line { x=2t y=-t z= -2t Is T invertible?
- 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.A) compute x the component of x that is parallel to L and x the component of x that is orthogonal to L B) compute the reflection of x across the line L; refL(x)More generally, if l is a line through the origin in R2, show that the transformation Pl : R2---> R2 that projects a point onto l is a linear transformation and find its standard matrix.Prove that Pl(cv) = cPl(v) for any scalar c
- Find the matrix [ T] C<--B of the linear transformation T : V ---> W with respect to the bases B and C of V and W, respectively. Verify by computing T(v) directly T: M22---> M22 defined by T (A ) = AT, B = C = {E11, E12, E21, E22}, v =AFind the matrix [ T] C<--B of the linear transformation T : V ---> W with respect to the bases B and C of V and W, respectively. Verify by computing T(v) directly T: P1--->P1 defined by T ( a + bx) = b -ax, B= {l + x, 1 -x}, C = {l, x}, v = p (x) = 4 + 2xSuppose T is a transformation from ℝ2 to ℝ2. Find the matrix A that induces T if T is a reflection over the line y=−4x.
- Find the matrix [ T] C<--B of the linear transformation T : V ---> W with respect to the bases B and C of V and W, respectively. Verify by computing T(v) directly T: P2--->R2 defined by T(p(x)) B={x2,x,1},C , v = p(x) =a + bx + cx2Suppose LMN->L'M'N'. If L(-8, 3) and L' is located at (-10, 8). What is the x value of the translation vector that maps LMN onto L'M'N'Find the matrix [ T] C<--B of the linear transformation T : V ---> W with respect to the bases B and C of V and W, respectively. Verify by computing T(v) directly T: P2-->R2 defined by T(p(x)) ,B = {l, x, x2}, C = {e1, e2}, v = p(x) = a + bx + cx2