Question

How to find nullity of a linear transformation?
For example find the nullity of the transformation
T(x,y)=(x,-y)

Answer 1

By nullity, do you mean the set of elements that are mapped to the zero element, or the set of elements which are unchanged by the transformation?

Answer 2

Set of elements that are mapped onto NULL element of the range
I need to find dimension of the set

Answer 3

Well that transformation is a reflection across the x-axis, so the only way you're getting the null element is if x = y = 0.

Answer 4

SO its dimension is one

Answer 5

No, it should be zero.

Answer 6

How did you determine the dimension of the null set

Answer 7

Well the null set is the zero vector. The plane, R2, is 2-dimensional. A line, R1, is one-dimensional. A point, in this case the point (0,0), is 0-dimensional.

Answer 8

So that means if we find a line on which each element is mapped onto O of range then the nullity would be 1 and for plane 2 and so on....

Answer 9

Yes, that's true. But if you are in R2, i.e. you have a 2x2 matrix, you'll never have a 2-dimensional nullspace because that would mean the entire plane is mapped to zero.

Answer 10

Unless of course you consider the transformation A(x,y) = (0,0) but that's kind of silly.

Answer 11

Thanks man

Answer 12

No problem