Linear maps

[munday63]

Hi, can someone explain to me how to show that the following is invertible?

Let L be L : R^2 -> R^2

Show that L(x,y) := (2x + y, 3x-5y) is linear and decide, with proof if L is invertible.

I cannot figure out how to prove it is linear and was wondering if someone could explain it to me.

[Glaysher]

Need to show L(ax₁ + bx₂, ay₁ + by₂)

= a L(x₁, y₁) + b L(x₂, y₂)

L(ax₁ + bx₂, ay₁ + by₂)

= (2(ax₁ + bx₂) + ay₁ + by₂, 3(ax₁ + bx₂) - 5(ay₁ + by₂))

=(2ax₁ + ay₁ + 2bx₂ + by₂, 3ax₁ - 5ay₁ + 3bx₂ - 5by₂)

= a(2x₁ + y₁, 3x₁ - 5y₁) + b(2x₂ + y₂, 3x₂ - 5y₂)

= a L(x₁, y₁) + b L(x₂, y₂)

Hence L linear