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The matrix is:

$$\begin{pmatrix} 3+l & 8 & 3 & 3+l \\ 8 & 9 & 3 & 7 \\ 3 & 3 & 7 & 8 \\ 3+l & 7 & 8 & 13 \end{pmatrix}$$

I'm given the above matrix, and I'm asked to figure if it can be the matrix of a linear transformation, for a given $l$.

What is the methodology to find a linear transformation that uses the above as its matrix?

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    Any matrix $A$ is the matrix of the linear transformation $v\mapsto Av$. Are you sure you have got your question right?2012-05-16
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    Maybe he meant a linear operator (automorphism) $V \to V$ thus $\det A$ should be nonzero.2012-05-16
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    the question is correct, but under what basis is A the matrix of v↦Av ? or what's the exact transformation that uses it?2012-05-16
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    Is this the correct transformation equation for say l=0? `f(x,y,z,w)=(3x+8y+3z+3w,8x+9y+3z+7w, 3x+3y+7z+8w, 3x+7y+8z+13w)`2012-05-16
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    Neyo, your answer's "almost" correct, but don't forget the "3+l", whatever is that "l", in the first and fourth lines, thus making it $\,((3+l)x+8y+3z+(3+l)z,\, etc....)$2012-05-16
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    I can choose l at will, and chose it to be 0 as i wrote2012-05-16

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