Find the Kernel of the Trace of a Matrix

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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
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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