Determine if the set S below is linearly independent in M2,3

Suppose that there exist constants a₁, a₂, a₃, a₄, and a₅ so that

Then, we have the matrix equality (Definition ME)

which yields the linear system of equations

−2a₁ + 4a₂ − a₃ − a₄ − a₅ = 0

3a₁ − 2a₂ − 2a₃ + a₄ + 2a₅ = 0

4a₁ + 2a₂ − 2a₃ − 2a₅ = 0

−a₁ + 2a₃ − a₄ = 0 3a₁ − a₂ + 2a₃ − a₅ = 0

−2a₁ + a₂ + 2a₃ + 2a₄ − 2a₅ = 0

By row-reducing the associated 6 × 5 homogeneous system, we see that the only solution is

a₁=a₂=a₃ = a₄ = a₅ = 0, so these matrices are a linearly independent subset of M_2,3.