Determine which subsets of a given set of vectors are linearly independent

Let

$A = (1,2), \quad B = (2,-4), \quad C = (2,-3), \quad D = (1,-2).$

Find all nonempty subsets of $\{ A,B,C,D \}$ which are linearly independent.

First, all of the one element subsets $\{ A \}, \{ B \}, \{ C \}, \{D \}$ are linearly independent.
Next, none of the three and four element subsets are linearly independent (any set of more than two vectors in $\mathbb{R}^2$ is dependent in $\mathbb{R}^2$ by Theorem 12.10).
So, it is left to check the pairs. We know from a previous exercise (Section 12.15, Exercise #6) that two vectors $(a,b)$ and $(c,d)$ in $\mathbb{R}^2$ are linearly independent if and only if $ad - bc\neq 0$ . So, then we have

$\begin{align*} \{ A,B \} \text{ is independent since } ad -bc = (1)(-4) - (2)(2) = -8 \neq 0 \\ \{ A,C \} \text{ is independent since } ad -bc = (1)(-3) - (2)(2) = -7 \neq 0 \\ \{ A,D \} \text{ is independent since } ad -bc = (1)(-2) - (2)(1) = -4 \neq 0 \\ \{ B,C \} \text{ is independent since } ad -bc = (2)(-3) - (-4)(2) = 2 \neq 0 \\ \{ B,D \} \text{ is dependent since } ad -bc = (2)(-2) - (-4)(1) = 0 \\ \{ C,D \} \text{ is independent since } ad -bc = (2)(-2) - (-3)(1) = -1 \neq 0. \end{align*}$

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Determine which subsets of a given set of vectors are linearly independent

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