determining subspaces

Question

determine if {A ∈ M_3x3 (R) : Tr(A) = 0 } is a subspace of M_3x3 (R)

Answer 1

It looks like you want to know if all real 3x3 matrices whose trace is zero also form a subspace of real 3x3 matrices.  Let us focus on closure under addition and scalar multiplication.

 

Addition:

 

Let A and B be real 3x3 matrices with Tr(A) = Tr(B) = 0

 

Is Tr(A + B) = 0 ?

 

Tr(A + B) = [(A₁₁ + B₁₁) + (A₂₂ + B₂₂) + (A₃₃ + B₃₃)] = (A₁₁ + A₂₂ + A₃₃) + (B₁₁ + B₂₂ + B₃₃)

 

= Tr(A) + Tr(B) = 0 + 0 = 0

 

There is closure under addition.

 

Scalar multiplication:

 

Let x ∈ R.  Is Tr(xA) = 0 ?

 

Tr(xA) = xA₁₁ + xA₂₂ + xA₃₃ = x(A₁₁ + A₂₂ + A₃₃) = xTr(A) = 0

 

There is closure under scalar multiplication.

 

You should be able to verify the other properties of a subspace are also satisfied.