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In Exercises $1-4,$ determine if the vectors are linearly independent. Justify each answer.$\left[\begin{array}{r}{1} \\ {-3}\end{array}\right],\left[\begin{array}{r}{-3} \\ {9}\end{array}\right]$

The vectors are linearly dependent.

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 7

Linear Independence

Introduction to Matrices

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In mathematics, the absolu…

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In Exercises $1-4,$ determ…

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in this video, we have two vectors, B one and B two, and we want the test. If they're linearly independent. Orly nearly dependent now What we could do is throw these both into a matrix and begin row reduction. But let's take a different method. This method works with just two vectors, which is our situation here. Notice that if we take and consider the first elements, they're only off by multiplication. By negative three. That makes us suspicious. Let's suppose we take negative three times V one. We would obtain negative three and positive nine. But that means negative three times V one is equal to V two, and in this case, negative three times V one minus V two is equal to the zero Vector, in other words, were able to find scale er's C one and C two not both. Zero such that C one B one plus C two V two equals zero vector by taking see one to be negative three and see to to be equal to negative one. This, then implies that the vectors or the set of vectors B one B two is linearly dependent. We have to be careful if we use this method because we're using just two vectors. And in this case, two vectors are always linearly dependent if they're multiples of each other like we just found.

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