3 3 1 Determine whether the following vectors are linearly independent in R2 three span R3, so if we add any vector(s) we create a linearly dependent set are to determine whether the functions cosx,1,sin2 (x/2) are linearly independent
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[PDF] Partial Solution Set, Leon §33 331 Determine whether the
3 3 1 Determine whether the following vectors are linearly independent in R2 three span R3, so if we add any vector(s) we create a linearly dependent set are to determine whether the functions cosx,1,sin2 (x/2) are linearly independent
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Partial Solution Set, Leonx3.3
3.3.1Determine whether the following vectors are linearly independent inR2.
(a) 2 1 ;3 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 ;1 3 ;2 4 . No. This can be shown in two ways. First, the easy way: If the span of the rst two vectors is all ofR2(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 matrixA. Then apply Gaussian elimination to show that there are nontrivial solutions to the homogeneous equationAx=0. (Recommended, if you feel that you need more practice.)3.3.2Same as (1), except that we are now inR3. 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 nd three vectors that are linearly independent (and we can), it is easy to show that those three spanR3, so if we add any vector(s) we create a linearly dependent set. In other words, a set of four vectors fromR3is inevitably linearly dependent. This leaves (a), (c), and (d). In each, let the vectors in question be the columns of a matrixA, and investigate the existence of nontrivial solutions toAx=0. The answers will be no in both (c) and (d), but yes in (a).3.3.3This is straightforward. A set containing only various multiples of a single nonzero
vector inR3(part (d), for example) generates a line through the origin. A set containing exactly two linearly independent vectors inR3(e.g., parts (c) and (e)) generates a plane through the origin, and a linearly independent set of three vectors inR3generates all of R3.3.3.4Now we're inR22, and things are less obvious at rst glance. (They were obvious before,
right?) Determine whether the following vectors are linearly independent inR22: (b) 1 0 0 1 ;0 1 0 0 ;0 0 1 0 . We must decide whether there is a nontrivial linear combination of these that produces the matrix of zeros. So consider a sum of the form a1 0 0 1 +b0 1 0 0 +c0 0 1 0 =a b c a For this to be identical to the zero matrix, we needa=b=c= 0, so only the trivial linear combination suces. These vectors (yes, they're matrices, but inR22these arevectors) are linearly independent inR22.3.3.8We are to determine whether a collection of vectors (polynomials, in this case) is linearly
independent inP3. (a) 1;x2, andx22 are linearly dependent, sincex22 = 1(x2)2(1). (c) Consider the solutions toa(x+ 2) +b(x+ 1) +c(x21) =0(here0is the zero polynomial, 0x2+0x+0). The resulting matrix equation can be written asAx=0, whereA=2 42 111 1 0