# Finding rank and nullity of a linear map.

## 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

## 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.

LCKurtz
Homework Helper
Gold Member

## 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?

Isn't the image 1 dimensional, being multiples of a?

but does the image include zero values?

LCKurtz
Homework Helper
Gold Member
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?

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

still the zero vector

LCKurtz
Homework Helper
Gold Member
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

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?

LCKurtz