# sol33 - Partial Solution Set Leon 3.3 3.3.1 Determine...

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Partial Solution Set, Leon § 3.3 3.3.1 Determine whether the following vectors are linearly independent in R 2 . (a) ± 2 1 ² , ± 3 2 ² . Yes. These are clearly not scalar multiples of one another, and when testing two vectors that’s all that we need to show. (c) ± - 2 1 ² , ± 1 3 ² , ± 2 4 ² . No. This can be shown in two ways. First, the easy way: If the span of the ﬁrst two vectors is all of R 2 (it is; they are linearly independent), all three cannot help being linearly dependent. Done. The almost-as-easy way: Set up a homogeneous system in which the three vectors in question are the columns of a matrix A . Then apply Gaussian elimination to show that there are nontrivial solutions to the homogeneous equation A x = 0 . (Recommended, if you feel that you need more practice.) 3.3.2 Same as (1), except that we are now in R 3 . A pair of vectors is linearly independent unless they are scalar multiples of one another, and that takes care of (e). In (b), even if we can ﬁnd three vectors that are linearly independent (and we can), it is easy to show that those three span R 3 , so if we add any vector(s) we create a linearly dependent set. In other words, a set of four vectors from

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sol33 - Partial Solution Set Leon 3.3 3.3.1 Determine...

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