If TA R2 rightarrow R2 is multiplication by A where A is giv

If T_A: R^2 rightarrow R^2 is multiplication by A where A is given by (-5 -1 10 2), then the range of T_A is all of R^2. True False Which of the following transformations in R^3 are one-to-one? Contraction with a factor of k Reflection about the xy-plane. orthogonal projection on the yz-plane. (a) Find the characteristic equation of the following matrix: [7 -2 -6 0 1 0 1 0 1] (lambda - 1)[lambda^2 + 8 lambda - 13] = 0 (lambda - 1)[lambda^2 - 8 lambda - 13] = 0 (lambda - 1)[lambda^2 + 8 lambda + 13] = 0 lambda^2 - 8 lambda + 13 = 0 (lambda - 1)[lambda^2 - 8 lambda + 13] = 0

Solution

1.If T_A:R²R² is multiplication by A, then the range of T_A, being equal to Col(A) is span{ (5,1)^T} and not R². The statement is False ( (10,2)^T = -2(-5,-1)^T).

2.a) contraction with a factor of k is one-to-one as it maps (x,y,z)^T to (kx,ky,kz)^T

b) Reflection about the xy plane is one-to-one as the reflection of the point

(x,y,z) across the xy plane is (x,y,-z).

c) Orthogonal projection on the yz plane is not one-to-one as it maps (x,y,z)^T to (0,y,z)^T ( it maps (x₁,y₁,z)^T and (x₂,y₁,z₁)^T to the same point (0,y₁,z₁)^T).

3. The characteristic equation of the matrix A is det(A- I₃)= 0 or, ³-9²+21-13= 0 or, (-1)(²-8+13)=0. The last option is the correct answer.