linear transformation

[kulchytska]

Suppose we are given some linear transformations T: R2 →R, J:R2 →R2 ,and K : R2→R2 .
(a) We may define a function F : R2 → R2 by setting F(x) = T(x)J(x). Determine whether or not F must be a linear transformation.
(b) We may also define a function G : R2 → R by setting G(x) = J(x) · K(x). Determine whether or not G must be a linear transformation.

Hi guys, please help me understand how can I know here whether this function should be a linear transformation? Do I have to prove that this function satisfies both rules of linear transformations:

1. T(x+y)=T(x)+T(y)T(x+y)=T(x)+T(y)
2. T(ax)=aT(x)T(ax)=aT(x)

or something else?

 

[Steven G]

What you wrote is correct if you wanted to show that T is a linear transformation. You however are being asked to show that F is a linear transformation.

In some textbooks TJ(x) = T[J(x)] while others define TJ(x) = J[T(x)]. Which way does your book define composition?

 

[Dr.Peterson]

Suppose we are given some linear transformations T: R2 →R, J:R2 →R2 ,and K : R2→R2 .
(a) We may define a function F : R2 → R2 by setting F(x) = T(x)J(x). Determine whether or not F must be a linear transformation.
(b) We may also define a function G : R2 → R by setting G(x) = J(x) · K(x). Determine whether or not G must be a linear transformation.

Hi guys, please help me understand how can I know here whether this function should be a linear transformation? Do I have to prove that this function satisfies both rules of linear transformations:

1. T(x+y)=T(x)+T(y)
2. T(ax)=aT(x)

or something else?

What you wrote is correct if you wanted to show that T is a linear transformation. You however are being asked to show that F is a linear transformation.

In some textbooks TJ(x) = T[J(x)] while others define TJ(x) = J[T(x)]. Which way does your book define composition?

As I read the problem, there is no mention of composition, only of multiplication of two functions (scalar multiplication and dot product, respectively).

@kulchytska, can you post an image of the actual problem?

But what Jomo said is true: You need to apply the rules listed (which I corrected above, as your pasting gave double copies of each) to the function F and G, not to T.