Determining linear independence by inspection
I was given a question asking me to "determine by inspection whether the following vectors are linear independent." I know how to determine if vectors are independent by putting in row-echelon form and looking for free columns, but I don't know if determining by inspection is referring to a different method.
Is anyone familiar with this? Is there a different way to determine linear independence?
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$\begingroup$"By inspection" literally means "by looking at it and seeing if there is anything obvious".
In the case of determining dependence of vectors, you know that the vectors are linearly dependent if you can write one of them as a linear combination of the others. So, for example, if you're given the vectors $x = (1, 0, 0)$, $y = (0, 1, 0)$ and $z = (4, 5, 0)$ then you can quickly see that $z = 4x + 5y$ and hence they are linearly dependent.
Similarly, if you see that the vectors are such that any attempt to combine them will result in non-zero values somewhere, then you can state that they are independent by inspection. Sometimes this might rely on knowing a few extra theorems - e.g. knowing that it takes $n$ linearly independent vectors to span $\mathbb{R}^n$, so if you can see that the group of vectors is able to span the space then they must be independent.
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