6. Let v₁ = 4 3 generated by {V₁, V₂, V3). V₂ = 8 [81 V3 = 6 5. [10] and u = 2. Deteremine if u is in the subsapce of R³ 2
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- Find the standard matrix for the reflection T of R3 in the line { x=2t y=-t z= -2t Is T invertible?Let A ∈ M nxn(F). Show that A is diagonalizable if and only if At isdiagonalizableSuppose T is the transformation from ℝ2 to ℝ2 that results from a reflection over the y-axis followed by a x-shear of 1. Find the matrix A that induces T. A=?
- 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 + cx2Find 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 + 2x
- show that f(z)=(z-1) cos(1/z+2) has an essential singularity at z=-2More 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 ca) Suppose T ∈ L(V ), m is a positive integer, and v ∈ V is such that Tm-1v does not equal 0 but Tmv=0. Prove that v, Tv, T2,........,T (m-1)v is linearly independent.