Question

1)Which of the following transformations are linear?

A. {y1=6x1,y2=5
B. {y1=2x1+x2,y2=2
C.{y1=2,y2=7,y3=9
D. {y1=x1+10,y2=x2
E. y1=−3x,y2=14x1,y3=8x1
F.y1=5x1−4x2+3x3,y2=10x2−2x3,y3=−9x1−7x2

Answer 1

are b,e,f are linear?

Answer 2

To test whether it is a linear transformation or not, by definition, we need to check
\[T(\alpha u + \beta v )= T(\alpha u)+ T(\beta v)\]
or give out a counter example
for a) let u =(1,2)
v = (0,1)
so, u+ v= (1,3)
T(u+v) = (6,5) \(\neq\)T(u) + T(v) = (6,5)+(0,5) = (6,10)
--> T is not a linear transformation
for b) let u =(1,2)
v= (0,1)
u+ v = (1,3)
T(u+v) = (2*1+3, 2)= (5,2)\(\neq\) T(u) + T(v) = (2*1,2)+(2*0+1)= (2,3)
for c) it is not a linear transformation and you have to prove it step by step
let u = (x1, x2,x3)
v = (t1, t2,t3)

\[T( u + v) = (2,7,9) \neq T( u) + T ( v)= ....\]

Answer 3

I may exagerate the problem, but I failed the course because of this stuff so that I am so careful when solving it.

Answer 4

I see so linearity jus means dos it follow principle of superposition

Answer 5

then if i use A.{y1=6x1,y2=5 from the question
T(y1,y2)=(6x1,5) do i plug in random numbers then use T(u+v)=T(u)+T(v) ?