Find the value(s) of $h$ for which the following set of vectors
\[\left \{ \mathbf{v}_1=\begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix}, \mathbf{v}_2=\begin{bmatrix}
h \\
1 \\
-h
\end{bmatrix}, \mathbf{v}_3=\begin{bmatrix}
1 \\
2h \\
3h+1
\end{bmatrix}\right\}\] is linearly independent.

(Boston College, Linear Algebra Midterm Exam Sample Problem)
 

We give two solutions. The first one uses the homogeneous system and the second one uses a determinant.

Solution 1.

Let us consider the linear combination
\[x_1\mathbf{v}_1+x_2\mathbf{v}_2+x_3\mathbf{v}_3=\mathbf{0}. \tag{*}\] If this homogeneous system has only zero solution $x_1=x_2=x_3=0$, then the vectors $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ are linearly independent.
We reduce the augmented matrix for the system as follows.
\[\left[\begin{array}{rrr|r}
1 & h & 1 & 0 \\
0 &1 & 2h & 0 \\
0 & -h & 3h+1 & 0
\end{array}\right] \xrightarrow{R_3+hR_2}
\left[\begin{array}{rrr|r}
1 & h & 1 & 0 \\
0 &1 & 2h & 0 \\
0 & 0 & 2h^2+3h+1 & 0
\end{array}\right].
\] From this, we see that the homogeneous system (*) has only the zero solution if and only if
\[2h^2+3h+1 \neq 0.\] Since we have
\[2h^2+3h+1=(2h+1)(h+1),\] if $h \neq -\frac{1}{2}, -1$, then $2h^2+3h+1 \neq 0$.

In summary, the vectors $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ are linearly independent for any $h$ except $-\frac{1}{2}, -1$.

Solution 2.

Note that the vectors $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ are linearly independent if and only the matrix
\[A:=[\mathbf{v}_1 \mathbf{v}_2 \mathbf{v}_3]=\begin{bmatrix}
1 & h & 1 \\
0 &1 &2h \\
0 & -h & 3h+1
\end{bmatrix}\] is nonsingular.

Also, the matrix $A$ is nonsingular if and only if the determinant $\det(A)$ is nonzero.
So we compute the determinant of the matrix $A$ by the first column cofactor expansion and obtain
\begin{align*}
\det(A)&=\begin{vmatrix}
1 & h & 1 \\
0 &1 &2h \\
0 & -h & 3h+1
\end{vmatrix}\\[6pt] &=\begin{vmatrix}
1 & 2h\\
-h& 3h+1
\end{vmatrix}\\[6pt] &=2h^2+3h+1\\
&=(2h+1)(h+1).
\end{align*}

Hence $\det(A)\neq 0$ if and only if $h\neq -\frac{1}{2}, -1$.
Therefore, the vectors $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ are linearly independent for any values of $h$ except $-\frac{1}{2}, -1$.

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