Question

Which of the following are linear transformations?

Choose Linear Not Linear  The function f:ℝ3→ℝ2 defined byf([x y z]^T)=[x−y 3y+z]^T.

Choose Linear Not Linear  The function a:ℝ→ℝ such that a(x)=(x−1)+(x−2)^2.

Choose Linear Not Linear  The function g:M2,2(ℝ)→M2,2(ℝ) defined by g(A)=2A+[1 2

3 4] Here, M2,2(ℝ)) is the vector space of 2×2matrices with real entries.

Choose Linear Not Linear  The function h:ℝ2→ℝ defined by h([xy])=x^2−y^2.

Answer 1

1. The function f:R³→R² is defined by f (x, y, z)^T= (x−y ,3y+z)^T. Let X₁ = (x₁, y₁, z₁)^T and X₂ = (x₂, y₂, z₂)^T be 2 arbitrary elements of R³ and let k be an arbitrary scalar. Then f(X₁+X₂) = f(x₁+x₂,y₁+y₂,z₁+z₂)^T = (x₁+x₂-y₁-y₂, 3y₁+3y₂+z₁+z₂)^T = (x₁−y₁ ,3y₁+z₁)^T+(x₂−y₂ ,3y₂+z₂)^T = f(X₁)+f(X₂). This implies that f preserves vector addition. Further, f(kX₁) = f(kx₁,ky₁, kz₁)^T = (kx₁−ky₁ ,3ky₁+kz₁)^T = k(x₁−y₁ ,3y₁+z₁)^T = kf(X₁). This implies that f preserves scalar multiplication. Hence f is a linear transformation.
2. The function a :R→R is defined by a(x)= (x−1)+(x−2)². Let x₁ and x₂ be 2 arbitrary elements of R. Then a(x₁+x₂) = (x₁+x₂-1)+(x₁+x₂-2)² ≠ (x₁−1)+(x₁−2)² +(x₂−1)+(x₂−2)² i.e. a(x₁+x₂) ≠ a(x₁)+a(x₂). This implies that a does not preserve vector addition. Hence a is not a linear transformation.
3. The function g: M_2,2 (R) → M_2,2 (R) is defined by g(A)=2A+B, where B =

                                                                                                                                         

1	2
3	4

Let A₁ and A₂ be 2 arbitrary elements of M_2,2 (R). Then g(A₁+A₂) = 2(A₁+A₂)+B 2A₁+B +2A₂+Bi.e. g(A₁+A₂)g(A₁)+g(A₂). This implies that g does not preserve vector addition. Hence g is not a linear transformation.

4. The function h: R²→R is defined by h(x,y)^T= x²−y². Let X₁ = (x₁,y₁)^T and X₂ = (x₂,y₂)^T be 2 arbitrary elements of R². Then h(X₁+X₂) = (x₁+x₂)²−(y₁+y₂)²≠ x₁²−y₁² + x₂²−y₂² i.e. h(X₁+X₂) ≠ h(X₁)+h(X₂). This implies that h does not preserve vector addition. Hence h is not a linear transformation.