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Linear transformations

  • Thread starter karnten07
  • Start date
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0
[SOLVED] Linear transformations

1. Homework Statement

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

a.) L: R^2[tex]\rightarrow[/tex]R^2,

(x1)
(x2)
[tex]\mapsto[/tex]
(2x1 + 3x2)
(0)

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

3. 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 im trying to say is, do i need to define a vector (y1, y2) aswell in the set of R^2?
 
Last edited:

Answers and Replies

213
0
For x,y [tex]\in[/tex]R^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
Science Advisor
Homework Helper
26,258
618
That's right. I think you've got it.
 

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