<s>[INST] How many function are there? Let $f_k \colon \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be a linear map defined by
$$ 
f_k \begin{pmatrix} 1 \\ 2 \\ k \end{pmatrix}=
\begin{pmatrix} 2+k \\ 3 \\ 0 \end{pmatrix},
\quad
f_k \begin{pmatrix} 2 \\ k+1 \\ -1 \end{pmatrix}=
\begin{pmatrix} 1 \\ 1 \\ -2 \end{pmatrix},
\quad
f_k \begin{pmatrix} -3 \\ 1 \\ 5 \end{pmatrix}=
\begin{pmatrix} 1 \\ k \\ 2 \end{pmatrix},
$$
How many functions $f_k$ are there?
 [/INST] First, consider the three vectors,
$$\left(\begin{array}{c}1\\2\\k\end{array}\right),\qquad \left(\begin{array}{c}2\\k+1\\-1\end{array}\right),\qquad \left(\begin{array}{c}-3\\1\\5\end{array}\right).$$
If they are linearly independent, then we can define a linear transformation from $\mathbb{R}^3$ to any vector space simply by saying what we want the value to be at those three vectors (since any specification of what happens to a basis gives a linear transformation). If the three vectors are linearly dependent, then any linear dependency between them must also be satisfied by their images.
So,  when are they linearly independent? The three vectors are linearly independent if and only if the determinant of the $3\times 3$ matrix that has them as columns is nonzero. Since
$$\begin{align*}
\det\left(\begin{array}{ccc}1 & 2 & -3\\
2 & k+1 & 1\\
k & -1 & 5
\end{array}\right) &= \left|\begin{array}{cc}k+1 & 1\\ -1 & 5\end{array}\right| -2\left|\begin{array}{cc}2 & 1\\k & 5\end{array}\right| -3\left|\begin{array}{cc}2 & k+1\\k & -1
\end{array}\right|\\
&= 5(k+1) + 1 -2\Bigl(10 - k\Bigr) -3\Bigl(-2 -k^2-k\Bigr)\\
&= 5k + 5 +1 - 20 + 2k +6 +3k^2 + 3k\\
&= 3k^2 +10k -8 = (k+4)(3k-2).
\end{align*}$$
So the three vectors are linearly independent if $(k+4)(3k-2)\neq 0$; that is, if $k\neq -4$ and $k\neq\frac{2}{3}$. In those cases, such a linear transformation exists, and since we've specified what happens at a basis, there is one and only one such linear transformation.
If $k=-4$, then the vectors are
$$\left(\begin{array}{r}1\\2\\-4\end{array}\right),\qquad \left(\begin{array}{c}2\\-3\\-1\end{array}\right),\qquad \left(\begin{array}{c}-3\\1\\5\end{array}\right).$$
Notce that if we add all three vectors we get $\mathbf{0}$. That means that we must have that when we add the three images we get $\mathbf{0}$ as well, otherwise, we don't have a linear transformation; this holds, since the images are
$$\left(\begin{array}{r}-2\\3\\0\end{array}\right),\qquad \left(\begin{array}{r}1\\1\\-2\end{array}\right),\qquad \left(\begin{array}{r}1\\-4\\2\end{array}\right).$$
In fact, this is (up to multiplying by a scalar) the only linear relation between the three vectors: they span a 2-dimensional subspace. Add a single vector to, say, the first two, to get a basis. Then you can define a linear transformation to be what we want at these three, and anything whatsoever on the third basis basis vector. Each choice will give you a linear transformation; so there are infinitely many possible linear transformations when $k=-4$.
Finally, when $k=\frac{2}{3}$, we have that the three vectors are
$$\left(\begin{array}{r}1\\2\\\frac{2}{3}\end{array}\right),\qquad \left(\begin{array}{r}2\\\frac{5}{3}\\-1\end{array}\right),\qquad \left(\begin{array}{r}-3\\1\\5\end{array}\right).$$
The three are linearly dependent; if you do a bit of work, you will find that
$$3\left(\begin{array}{c}1\\2\\ \frac{2}{3}\end{array}\right) - 3\left(\begin{array}{r}2\\ \frac{5}{3}\\-1\end{array}\right) -\left(\begin{array}{r}-3\\1\\5\end{array}\right) = \left(\begin{array}{c}3 -6 +3\\ 6-5-1\\2+3-5
\end{array}\right) = \left(\begin{array}{c}0\\0\\0\end{array}\right).$$
That means that we need the images to satisfy the same linear relation; but
$$3\left(\begin{array}{c}2+\frac{2}{3}\\3\\0\end{array}\right) -3\left(\begin{array}{c}1\\1\\-2\end{array}\right) - \left(\begin{array}{c}1\\\frac{2}{3}\\2\end{array}\right)\neq\left(\begin{array}{c}0\\0\\0\end{array}\right),$$
so there can be no linear transformation with those values when $k=\frac{2}{3}$.
a
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