Finding rank and nullity of a linear map.

  • Thread starter sg001
  • Start date
  • #1
134
0

Homework Statement




let a be the vector [2,3,1] in R3 and let T:R3-->R3 be the map given by T(x) =(ax)a

State with reasons, the rank and nullity of T


Homework Equations





The Attempt at a Solution



Im having trouble understanding this... I know how to do this with a matrix ie row reduce and no. of leading cols = rank ,, and then no. of non leading cols = nullity.

But im stuck on how to go about it with this eqn.?

And that the rank = dim(image)

but how would I finad that...

or alternatively the dim(kernal)?

Thanks.
 

Answers and Replies

  • #2
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
9,559
773

Homework Statement




let a be the vector [2,3,1] in R3 and let T:R3-->R3 be the map given by T(x) =(ax)a

State with reasons, the rank and nullity of T

Isn't the image 1 dimensional, being multiples of a?
 
  • #3
134
0
Isn't the image 1 dimensional, being multiples of a?

but does the image include zero values?
 
  • #4
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
9,559
773
but does the image include zero values?

Do you mean the zero vector? If ##x = \theta##, the zero vector, what is ##T(\theta)## by your formula?
 
  • #5
134
0
Do you mean the zero vector? If ##x = \theta##, the zero vector, what is ##T(\theta)## by your formula?

still the zero vector
 
  • #6
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
9,559
773
but does the image include zero values?

Do you mean the zero vector? If ##x = \theta##, the zero vector, what is ##T(\theta)## by your formula?

still the zero vector

Does that answer your question?
 
  • #7
134
0
Does that answer your question?

yes, so the image cointains the zero value... but I thought the image was the set of all function values except 0, I thought that was the kernal

or is it that the kernal is a proper subset of the image?
 
  • #8
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
9,559
773
yes, so the image cointains the zero value... but I thought the image was the set of all function values except 0, I thought that was the kernal

or is it that the kernal is a proper subset of the image?

At this point, I suggest you look up the definition of the kernel of a linear transformation in your text.
 

Related Threads on Finding rank and nullity of a linear map.

  • Last Post
Replies
13
Views
3K
  • Last Post
Replies
4
Views
1K
Replies
5
Views
4K
  • Last Post
Replies
7
Views
2K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
2
Views
2K
Replies
4
Views
2K
Replies
11
Views
5K
Replies
1
Views
3K
Top