Linear tranformations equality

[sphlanx]

Homework Statement

We are given a homogeneous system of 5 equations and 3 variables. We are asked to find the solutions (which i found to be unique and <0,0,0>) and then we are asked(along loads of other stuff:P):

a)If any two linear transformations g and f have kerf=kerg=<0,0,0> then they are equal.
b)If any two linear transformations g and f have imf=img=<0,0,0> then they are equal.

Homework Equations

The Attempt at a Solution

a) is wrong because there are infinite linear transformations with ker=<0,0,0> but different images. for (positive but not absolutely sure i am correct because the dimension of the images would be equal)
i think b) is correct but i am not sure. I would say that if imf=img=<0,0,0> then they are both the zero linear map so they are equal. Thanks again in advance

 

[rochfor1]

Sounds to me like you've got it down!

 

[Architeuthis Dux]

I would first commend you on finding the unique solution to the system of equations and for recognizing the importance of the question being asked. Your reasoning for part a) is correct - having the same kernel does not necessarily mean that two linear transformations are equal because there can be infinite transformations with the same kernel but different images. However, for part b), I would suggest that you provide a more thorough explanation. While it is true that if imf=img=<0,0,0> then they are both the zero linear map, it is important to also mention that the zero linear map is unique and therefore, any two transformations with the same image must be equal to the zero linear map and, by extension, equal to each other. This is because the image of a linear transformation is determined solely by its action on the basis vectors of the vector space, and if both transformations have the same image, they must have the same action on the basis vectors and therefore, be equal. Overall, your understanding and explanation of part b) is correct.