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

Specific Linear Map Example

  • Thread starter *melinda*
  • Start date
86
0
Homework Statement
Give a specific example of an operator T on R^4 such that,

1. dim(nullT) = dim(rangeT) and

2. dim(the intersection of nullT and rangeT) = 1

The attempt at a solution
I know that dim(R^4) = dim(nullT) + dim(rangeT) = 4, so dim(nullT) = dim(rangeT) = 2.

I also know that nullT will have 2 basis vectors and rangeT will also have 2 basis vectors (so that 1. is satisfied), and that they must have one vector in common (so that 2. is satisfied).

I started with T(w, x, y, z) = (w, x, 0, 0), but that only does it halfway.
I just don't know how to generate an example that satisfies both conditions.
Any ideas?
 

Dick

Science Advisor
Homework Helper
26,258
618
Try T(w,x,y,z)=(w,0,x,0). This is somewhat confusing because you usually think of null(T) and range(T) as living in different spaces. Write it in terms of a basis {e1,e2,e2,e4}. T(e1)=e1, T(e2)=0, T(e3)=e2, T(e4)=0. So null(T)=span(e2,e4). range(T)=span(e1,e2).
 

Related Threads for: Specific Linear Map Example

Replies
1
Views
749
Replies
6
Views
2K
  • Posted
Replies
3
Views
851
  • Posted
Replies
0
Views
2K
  • Posted
Replies
2
Views
1K
  • Posted
Replies
0
Views
703
  • Posted
Replies
9
Views
3K
  • Posted
Replies
1
Views
1K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top