Let T : R n → R m be a linear transformation, and let {v1, v2, v3} be a linearly dependent set in R n. Prove that the set {T(v1), T(v2), T(v3)} is also linearly dependent.

Answer 1

Given that exist C₁v₁ + C₂v₂ + C₃v₃ = 0, such that not all from C₁, C_2,, C₃ equel 0.

Then T(C₁v₁ + C₂v₂ + C₃v₃) = C₁T(v₁) + C₂T(v₂) + C₃T(v₃) = T(0) = 0.

So, {T(v₁), T(v₂), T(v₃)} is linearly dependent, because not all from C₁, C_2,, C₃