Find the matrix of the linear transformation 𝑇:ℝ3→ℝ3 which projects a vector onto the plane 𝑥−𝑦+2𝑧=0.

Answer 1

Normal vector to this plane is N = <1, - 1, 2> Let we take any vector M = <x, y, z>.

Then projection vector of M on given plane is P = M - [(M•N)/(N•N)]N = <x, y, z> - (x - y + 2z)/6·• <1, - 1, 2> =

<x - (x - y + 2z)/6, y + (x - y + 2z)/6, z - (x - y + 2z)/3 > = <(5x + y - 2z)/6, (x + 5y + 2z)/6, (- x + y + z) /3>.

Now we can find images of ortogonal basic in R3.

T(e₁)= T(<1, 0, 0>) = <5/6,1/6, - 1/3>; T(e₂) = T(<0, 1, 0>)= <1/6, 5/6, 1/3>

T(e₃) = T(<0, 0, 1>) = <- 1/3, 1/3, 1/3>. I5/6 1/6 -1/3I

So matrix A of this linear transformation is A = I1/6 5/6 1/3I

I-1/3 1/3 1/3I