In problems 110 a transformation T from one Euclidean space

In problems 1-10, a transformation T from one Euclidean space to another is defined. Determine whether or not T is a linear transformation. If so, find the matrix A such that T(x)=Ax. If not tell why not

5) T(x,y,z)=(x,y^2,z^3)

9) T(x,y,z)=(2x-3y+4z, 3x-5y-7z)

Please show steps

Solution

Let T : V W be a mapping.

5) Let a be an arbitrary scalar and let u = (x,y,z) be an arbitrary element of V. Then T(au) = T(ax,ay,az) = (ax, a²y² , a³z³) and aT(u) = a(x,y²,z³) = (ax, ay² , az³ ). Thus, T(au) aT(u). Thus, T does not preserve scalar multiplication. Hence T is not a linear transformation.

9) Let a be an arbitrary scalar and let u = ((x₁,y₁,z₁)) and v = (x₂ ,y₂,z₂) be two arbitrary elements of V. Then T(u+v) = T (x₁+x₂, y₁+y₂, z₁+z₂)   = ( 2(x₁+x₂) -3(y₁+y₂) +4(z₁+z₂),   3(x₁+x₂) -5(y₁+y₂) -7(z₁+z₂)) = (2x₁-3y₁+4z₁, 3x₁-5y₁-7z₁) + (2x₂-3y₂+4z₂ , 3x₂-5y₂-7z₂) = T(u)+T(v). Hence T preserves vector addition.

Further T(au) = T(ax₁ ,ay₁ ,az₁) = ( 2ax₁-3ay₁ +4az₁ , 3ax₁-5ay₁-7az₁) = a( 2x₁- 3y₁+4z₁, 3x₁-5y₁-7z₁) = aT(u). Hence T preserves scalar multiplication also. Thus T is a linear transformation.