Decide if this subset is a basis of the vector space

Question

Decide if the subset is a basis of the vector space $V$.

$\left\{v_{1},v_{2},v_{3}\right\} = \left\{ \begin{pmatrix} 1\\ 2\\ 1 \end{pmatrix},\begin{pmatrix} -1\\ 3\\ 2 \end{pmatrix},\begin{pmatrix} 3\\ 1\\ -2 \end{pmatrix} \right\}$ in $V = \mathbb{R}^{3}$

I think an equivalent question would be "Are these three vectors linearly independent?".

So I would write these three vectors as a matrix and then use Sarrus rule to calculate the determinant. If determinant $\neq 0$, then the subset is a basis of the vector space $V$.

$$\begin{vmatrix} 1 & -1 & 3\\ 2 & 3 & 1\\ 1 & 2 & -2 \end{vmatrix} \begin{matrix} 1 & -1\\ 2 & 3\\ 1 & 2 \end{matrix}$$

$$\text{Determinant} = -10$$

Thus the three vectors are linearly independent and this means the subset $\left\{v_{1},v_{2},v_{3}\right\}$ is a basis of the vector space $V$.

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Please tell me if I did it correctly and if not how to do it correctly?

Answer 1

This is correct. The definition of linearly independent is, for $v_1,v_2,v_3 \in V = \mathbb{R}^3$ we have that, for $\alpha,\beta,\gamma \in \mathbb{R}$ these vectors are linearly independent if the relation below is true

$$\alpha v_1 + \beta v_2 + \gamma v_3 = 0 \iff \alpha = \beta = \gamma = 0$$

So, for the vectors pointed out by you in your question you have that

$$\left\{\begin{matrix} \alpha - \beta + 3\gamma=0\\ 2\alpha + 3\beta + \gamma=0\\ \alpha + 2\beta - 2\gamma=0 \end{matrix}\right.$$

Solving this system is related to set the matrix

$$\left(\begin{matrix} 1 & -1 & 3\\ 2 & 3 & 1\\ 1 & 2 & -2 \end{matrix}\right)$$

in row echelon form. Using linear transformations we get the matrix in the form

$$\left(\begin{matrix} 1 & -1 & 3\\ 0 & -5 & 5\\ 0 & 0 & -2 \end{matrix}\right)$$

Which is related to the system

$$\left\{\begin{matrix} \alpha - \beta + 3\gamma=0\\ -5\beta + 5\gamma=0\\ - 2\gamma=0 \end{matrix}\right.$$

That has only the trivial solution. So we can conclude that these vectors are linearly independent. This problem is equivalent to search for the determinant as you did above because it is true that for a Homogeneous system of equations that have a related matrix, let's say the $A$ matrix, have only the trivial solution when the determinant is different then zero.