Determine whether T R2 rightarrow R defined by Tx1 x2 3x1

Determine whether T: R^2 rightarrow R defined by T(x_1, x_2) = 3x_1 - 2x_2 is a linear transformation.

Solution

we need to check whether T : R² R define by T(x₁,x₂) = 3x₁ - 2x₂ is linear transformation or not.

we can say that T is linear transformation if it satisfies vector addition and scalar multiplication that is :

1) T(x+y) = T(x) + T(y) 2) T(cx) = cT(x)

we have,

T(x₁,x₂) = 3x₁ - 2x₂

we can write,

T(x) = 3x₁ - 2x₂   --------------------------------1)

T(y) = 3y₁ - 2y₂ ------------------------------2)

check the vector addition property we have,

T(x+y) = T(x₁+y₁ , x₂+y₂)

T(x+y) = 3(x₁+y₁) - 2(x₂+y₂)

T(x+y) = 3x₁ + 3y₁ - 2x₂ - 2y₂

T(x+y) = 3x₁ - 2x₂ + 3y₁ - 2y₂

from equation 1) and 2) we can say that,

T(x+y) = T(x) + T(y)

check the scalar multiplication property

we have,

T(x) = 3x₁ - 2x₂

T(cx) = T(cx₁,cx₂)

T(cx) = 3cx₁ - 2cx₂

T(cx) = c(3x₁ - 2x₂)

T(cx) = cT(x)

hence we can say that T satisfies both the properties of vector addition and scalar multiplication.

we can say that T is a linear transformation.