Find the Kernel of the Trace of a Matrix

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The discussion focuses on determining the kernel and image of the linear transformation F, defined as the trace of a matrix. It is established that F is indeed a linear transformation. The kernel of F consists of all matrices A in Mnn such that tr(A) = 0, while the image of F is the set of all real numbers, ℝ. Clarification is sought on the distinction between the kernel as a set of matrices and the image as a set of numbers. The conversation emphasizes understanding the mapping of matrices to their traces and the implications for kernel and image.
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Homework Statement



Let F : Mnn(R) → R where F(A) =tr(A). Show that F is a linear transformation. Find the kernel of F as well as its dimension. What is the image of F?


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The Attempt at a Solution



I have shown that it is a linear transformation. But I am not sure about the Ker(F) and Im(F),
would Ker(F) just be {tr(A), for A in Mnn}? And would the Im(F) just be {a : a\inℝ}? Thanks.
 
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Do you understand what the problem is asking? You are gvien a linear transformation that maps every matrix to a number, its trace. This problem is asking for the trace of that linear transforamation- the set of matrices that are mapped to 0. It is asking for a set of matrices, not a set of numbers.
 
Okay, so would I say kernel is {A where tr(A)=0, for A in Mnn}? So what would the image be then? Thanks
 

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