Given a matrix, find a linear transformation that uses it

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? – Chris Eagle May 16 '12 at 10:58
Maybe he meant a linear operator (automorphism) $V \to V$ thus $\det A$ should be nonzero. – Yrogirg May 16 '12 at 11:01
the question is correct, but under what basis is A the matrix of v↦Av ? or what's the exact transformation that uses it? – Neyo May 16 '12 at 11:19
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) – Neyo May 16 '12 at 11:28
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 $\,((3+l)x+8y+3z+(3+l)z,\, etc....)$ – DonAntonio May 16 '12 at 11:36

1 Answer

As commentators have mentioned, any matrix produces a linear transformation on vectors through multiplication. In detail using $A$ for your matrix, $[w,x,y,z]A=[(8 x+w (3+x)+3 y+(3+x) z, 8 w+9 x+3 y+7 z, 3 w+3 x+7 y+8 z, 7 x+w (3+x)+8 y+13 z)]$

In fact, the same matrix can represent different transformations depending on the basis being used. If you are certain you are supposed to determine if this is a matrix for a given transformation, then you will need to add information about the basis.

In the case Yrogirg is correct about you wanting it to be a nonsingular transformation, then the course of action would be to compute the determinant and see if/when it is zero. You should get $-54\ell^2+501\ell-1167$. By checking the disriminant you can see that it only has two complex roots, so this matrix is always nonsingular (if you are only interested in real matrices.)

I noticed that the matrix is also symmetric, which I thought might come into the picture somehow.

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The question is if this is a matrix for any linear transformation, meaning I just have to find one myself. It was simpler than I thought I guess. (Yes the symmetric part helps with other questions about diagnolization, already noted.) – Neyo May 16 '12 at 11:48
One thing before I accept the answer: Is there something that guarantees that f (the one i wrote above) is linear, without running and proving the properties of linear equations (meaning the f(a+b)=f(a)+f(b) and f(l*a)=lf(a))? – Neyo May 16 '12 at 11:53
Any transformation you write that way (making combinations with only addition, subtraction, scalars, and first degree powers of the inputs) is a linear transformation. Convince yourself by randomly generating such a transformation, and checking its linearity. Then maybe make a matrix for your transformation for practice! – rschwieb May 16 '12 at 11:56
Frankly speaking, I still don't understand what Neyo wanted. Neyo, did you solve the problem? If so, post the answer, so people like me could understand the question. – Yrogirg May 17 '12 at 19:00