• Support PF! Buy your school textbooks, materials and every day products Here!

Matrix representation for a transformation

  • Thread starter fk378
  • Start date
  • #1
367
0

Homework Statement


Consider the linear transformation T: R2-->R2, where
T(x1, x2, x3)= (3x2-x3, x1+4x2+x3)

a. Find a matrix which implements this mapping.
b. Is this transformation one-to-one? Is it onto? Explain.

Homework Equations


[T(x)]_B = ([T]_B) (x_B)


The Attempt at a Solution


The matrix that implements this mapping would be the representation ([T]_B). I think that (x_B) is the vector [x1, x2, x3] and that [T(x)]_B is (3x2-x3, x1+4x2+x3) relative to the {x1, x2, x3} basis. So then ([T]_B) must be the matrix:

0 3 -1
1 4 1

Row-reducing this matrix to echelon form gives 2 pivots, so the transformation is onto since the system is consistent, but it is not one-to-one because the system is linearly dependent.


Are all my thoughts correct for this problem?
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
618
The matrix is correct. T:R3->R2, not R2->R2. Your conclusions are also correct. Though I would probably figure them out in a less abstract way than counting 'pivots'.
 
  • #3
367
0
How would you come to the conclusions? Would you first just point out that there are more columns than rows so one-to-one is not a possibility?
 
  • #4
HallsofIvy
Science Advisor
Homework Helper
41,792
919
You might, for instance, find two distinct vectors of R3 which map into the same thing.
 

Related Threads for: Matrix representation for a transformation

Replies
0
Views
1K
Replies
1
Views
1K
Replies
4
Views
629
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
9
Views
3K
  • Last Post
Replies
3
Views
3K
Top