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user100 [1]
3 years ago
12

In this exercise, T: R2 → R2 is a function. For each of the following parts, state why T is not linear. (a) T(a1, a2)= (1, a2) (

b) T(a, a2) (a1,ai) (c) T(a1,a2)= (sin a1,0) (d) T(al, aa) = (lal, a2) (e) T(a1, d12) = (a1 + 1, a2)
Mathematics
1 answer:
DENIUS [597]3 years ago
8 0

Answer:

  • a) T is not homogeneous
  • b) T is not additive
  • c) T is not homogeneous
  • d) T is not additive
  • e) T is not additive

Step-by-step explanation:

In each case there is an example where one property of linear functions fails.

  • a) T(2(1,0))=T((2,0))=(1,0); 2T((1,0))=2(1,0)=(2,0). These vectors are not equal, then T doesn't satisfy the condition of scalar multiplication (homogeneity).
  • b) T((1,2)+(2,3))=T(3,5)=(3,9); T((1,2))+T((2,3))=(1,1)+(2,4)=(3,5). Because these vectors are not equal, T doesn't satisfy the property of vector addition (additivity).
  • c) T(\frac{1}{2}(\pi,0))=T((\frac{\pi}{2},0))=(\sin(\frac{\pi}{2}),0)=(1,0); \frac{1}{2}T((\pi,0))=\frac{1}{2}(0,0)=(0,0) so T is not homogeneous.
  • d) T((-1,0)+(1,0))=T(0,0)=(0,0); T((-1,0))+T((1,0))=(1,0)+(1,0)=(2,0) then T is not additive.
  • e) T((0,0)+(1,0))=T(1,0)=(2,0); T((0,0))+T((1,0))=(1,0)+(2,0)=(3,0) then T is not additive.
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