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proving a linear transformation is an isomorphism
Let $T:\mathbb{R}^3\to \mathbb{R}^3$ be defined by $T(x,y,z)=(3z,z+y,x-y)$.
I am asked to prove that $T$ is an isomorphism.
I have used the rank-nullity theorem several times and the fact that $dim\ R^3$ is $3$, but I have no idea how to continue.
A:
The statement says $T$ is injective and surjective. Injectivity is clear. To show surjectivity, you need to show that for any $x\in\mathbb R^3$ there exists a $v\in\mathbb R^3$ such that $T(v)=x$.
The vector $v$ can be written as
$$v=a(1,0,0)+b(0,1,0)+c(0,0,1)$$
where $a,b,c\in\mathbb R$. Then
$$T(v)=(3c,c+a,a-b)$$
and $T(v)=x$ is the statement.
Q:
Visual Studio 2010 Blank Project Error
I started a new project in Visual Studio 2010 using ‘Windows Forms Application’ template. The project immediately gave error which says that:
“This project file contains one or more targets that are not valid for this version of Visual Studio. The targets which are not valid for this version are: “.
I tried to export the project from VS 2010 to Visual Studio 2008, but the problem still persists.
Has anyone faced this issue before?
A:
It’s an old bug, go to Tools -> Options -> Projects and Solutions -> Installed -> uncheck the box for ‘Visual Studio 2008 project types’.
A:
This turned out to be a file association issue. I had to manually change the file type association in the file system, it was associated with a.bmp file and if I change it to a.jpg file then it
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