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

Linear transformations

  • Thread starter oomba
  • Start date
  • #1
2
0

Homework Statement



T:[tex]{R^3 \rightarrow {R^2}[/tex] given by [tex]T(v_1,v_2,v_3) = (v_3 -v_1, v_3 - v_2)[/tex]

If linear, specify the range of T and kernel T


The attempt at a solution
Okay, I went ahead and tried to find the kernel of T like here:
[tex]\begin{align*}&v_3 - v_1 = 0\\
&v_3 - v_2 = 0\end{align*}[/tex]

Thus, [tex]\begin{align*}&v_3 = v_1 \\
&v_3 = v_2\end{align*}[/tex]

So choosing v3 as s gives the 1-D basis of W= s(1, 1, 1) **a column vector**

But I'm not entirely sure how to get the range. IF I did the kernel correctly, then that means the dimension of the range will be 2 as 2+1 = 3 (the dimension of the domain). But when I try to do the range, I get a 3-dimensional basis where v1,v2,and v3 are their own LI vectors as so:
(y1,y2) = s(1,1) + t(-1,0) + r(0,-1)

Any help?
 

Answers and Replies

  • #2
354
0
Everything seems fine. So where's your difficulty?
 
  • #3
HallsofIvy
Science Advisor
Homework Helper
41,793
922
You have correctly deduced that the range must have dimension 2 and you know that the range is a subspace of R2.

How many subspaces of dimension 2 do you think R2 has!
 

Related Threads for: Linear transformations

  • Last Post
Replies
1
Views
701
  • Last Post
Replies
7
Views
869
  • Last Post
Replies
8
Views
2K
  • Last Post
Replies
6
Views
1K
  • Last Post
Replies
6
Views
853
  • Last Post
Replies
4
Views
761
  • Last Post
Replies
9
Views
560
  • Last Post
Replies
10
Views
2K
Top