Let T R2 rightarrow R3 be a function defined by Tx y x y x

Let T: R^2 rightarrow R^3 be a function defined by T(x, y) = (x = y, x + y, y) Show that T is a linear transformation.

Solution

For T is a linear transformation, we need to show if a = (a₁, a₂) and b = (b₁,b₂)

then, T(a+b) = T(a) + T(b)

and for any scalar s. we have T(sa)= sT(a).

T(a+b) = T ((a₁,a₂)+(b₁,b₂)) = T (a₁ + b₁, a₂ + b₂)

= <a₁ + b₁ = a₂ + b₂, a₁ + b₁ + a₂ + b_2, a₂ + b₂>

= <a₁ = a₂, a₁ + a₂ , a₂> + < b₁ = b₂, b₁ + b_2,, b₂>

= T(a₁,a₂) + T(b₁,b₂)

= T(a) + T(b)

Also, T(sa) = T (s(a₁,a₂)

= T (sa₁,sa₂)

= <sa₁ = sa₂, sa₁ + sa₂, sa₂>

= s<a₁ = a₂, a₁ + a₂, a₂>

= sT(a₁,a₂)

= sT(a)

Therefore, T is a linear transformation.