Answer to Question #264358 in Linear Algebra for Alish

Question

 In each of the following cases explain whether R²$\to$ R is a linear transformation, if it is, supply a proof, if not, supply a counter ample

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

b) T(a, b) =ab

c) T(a, b) =|a|²

d) T(a, b) =a - b

Answer 1

for linear transformation:

$T(a+b)=T(a)+T(b)$

$T(ka)=kT(a)$

a)

$T((a_1,b_1)+(a_2,b_2))=T((a_1+a_2),(b_1+b_2))=a_1+a_2+b_1+b_2$

$T(a_1,b_1)+T(a_2,b_2)=a_1+b_1+a_2+b_2$

$T(k(a,b))=ka+kb=k(a+b)=kT(a,b)$

this is a linear transformation

b)

$T((a_1,b_1)+(a_2,b_2))=T((a_1+a_2),(b_1+b_2))=(a_1+a_2),(b_1+b_2)$

$T(a_1,b_1)+T(a_2,b_2)=a_1b_1+a_2b_2$

this is not a linear transformation

for example: $a_1=1,a_2=2,b_1=3,b_2=4$

$T((a_1,b_1)+(a_2,b_2))=21\neq T(a_1,b_1)+T(a_2,b_2)=11$

c)

$T((a_1,b_1)+(a_2,b_2))=T((a_1+a_2),(b_1+b_2))=|a_1+a_2|^2$

$T(a_1,b_1)+T(a_2,b_2)=|a_1|^2+|a_2|^2$

this is not a linear transformation

for example: $a_1=1,a_2=2,b_1=3,b_2=4$

$T((a_1,b_1)+(a_2,b_2))=9\neq T(a_1,b_1)+T(a_2,b_2)=5$

d)

$T((a_1,b_1)+(a_2,b_2))=T((a_1+a_2),(b_1+b_2))=a_1+a_2-b_1-b_2$

$T(a_1,b_1)+T(a_2,b_2)=a_1-b_1+a_2-b_2$

$T(k(a,b))=ka-kb=k(a-b)=kT(a,b)$

this is a linear transformation