Linear Transformations: Proofs and Examples for R^2 to R^2

[karnten07]

[SOLVED] Linear transformations

Homework Statement

Determine whether the following maps are linear transformations. (proofs or counterexamples required)

a.) L: R^2\rightarrowR^2,

(x1)
(x2)
\mapsto
(2x1 + 3x2)
(0)

The brackets should be two large brackets surrounding the two vectors.

The Attempt at a Solution

I've been reading about linear transformations and i know i have to show something like:

L(x1+x2)= L(x1) +L(x2) and L(cx1)= cL(x1) where c is a scalar.

Is this right and i should treat x1 and x2 separately rather than the vector including x1 and x2 as one element of R^2?

What I am trying to say is, do i need to define a vector (y1, y2) as well in the set of R^2?

 

[karnten07]

For x,y \inR^2,

x=(x1,x2) and y= (y1,y2)

x+y=(x1,x2)+(y1,y2)= (x1+y1, x2+y2)
L(x+y) = L(x1+y1, x2+y2)
=2(x1+y1)+3(x2+y2)
=(2x1+3x2)+(2y1+3y2)
=L(x) +L(y)

Is this right for the first part?

Then because cx= c(x1,x2) = (cx1,cx2) you have
L(cx) = L(cx1, cx2) = (2cx1 + 3cx2)
=c(2x1 + 3x2)
=cL(x)

I think I'm understanding it more now, if this is right that is.

 

[Dick]

That's right. I think you've got it.