Determine if Linear S[[x],[y],[z]]=(x+y+z)*[[1],[z]]

Problem

S*[[x],[y],[z]]=(x+y+z)*[[1],[z]]

Solution

1. Identify the definition of a linear transformation. A transformation S is linear if it satisfies S*(u+v)=S(u)+S(v) (additivity) and S*(c*u)=c*S(u) (homogeneity) for all vectors u,v and scalars c

2. Expand the expression for the transformation to see the components of the output vector.

S*[[x],[y],[z]]=[[x+y+z],[x*z+y*z+z2]]

3. Test for homogeneity by multiplying the input vector by a scalar c

S*(c*[[x],[y],[z]])=S*[[c*x],[c*y],[c*z]]

4. Substitute the scaled components into the transformation formula.

S*[[c*x],[c*y],[c*z]]=(c*x+c*y+c*z)*[[1],[c*z]]

S*[[c*x],[c*y],[c*z]]=[[c*(x+y+z)],[c2*(x*z+y*z+z2)]]

5. Compare the result to c*S(x,y,z)

c*S*[[x],[y],[z]]=c*[[x+y+z],[x*z+y*z+z2]]

c*S*[[x],[y],[z]]=[[c*(x+y+z)],[c*(x*z+y*z+z2)]]

6. Conclude that since the second component involves c2 in the first calculation and c in the second, S*(c*u)≠c*S(u) for c≠1,0 The presence of the z2 x*z and y*z terms indicates the transformation is non-linear.

Final Answer

S*[[x],[y],[z]]=(x+y+z)*[[1],[z]]* is not linear

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