- Thread starter
- #1

- Feb 5, 2012

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Here is a question I encountered recently.

**Question:**

Let \(V\) be a unitary space. Give the definitions of a self adjoint and unitary linear transformations of \(V\). Prove that \(f_1 g_1=f_2 g_2\), where \(f_1,\,f_2\) are self adjoint positive and \(g_1,\,g_2\) unitary implies \(f_1=f_2,\, g_1=g_2\) as soon as all transformations are non singular.

**My Answer:**

I know the definitions of the self adjoint and unitary linear transformations. As we have been taught in class they are as follows.

Let \(f:V\rightarrow V\) be a linear transformation and \((.\,,\,.)\) denote the associated Bilinear Form. Then \(f\) is called self adjoint if, \((f(x),\,y)=(x,\,f(y))\) for all \(x,\,y\in V\). Similarly \(f\) is called unitary if \((f(x),\,f(y))=(x,\,y)\) for all \(x,\,y \in V\).

Now the problem I have is how to tackle the second part of the question. If anybody could give me a hint on how to proceed that would be really nice.