# 235HW5 - Math 235 Linear Algebra HW 5 Problem 2.6.2 Problem...

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Unformatted text preview: Math 235: Linear Algebra HW 5 Problem 2.6.2 Problem 2.6.2 Let α 1 = (1 , 1 ,- 2 , 1) , α 2 = (3 , , 4 ,- 1) , α 3 = (- 1 , 2 , 5 , 2) and Let α = (4 ,- 5 , 9 , 7) , β = (3 , 1 ,- 4 , 4) , γ = (- 1 , 1 , , 1) (a) Which of the vectors α , β , γ are in the subspace of R 4 spanned by the α i ? Put the α i into a matrix and row-reduce. 1 1- 1 1 3 4- 1- 1 2 5 2 → 1 0 0- 3 / 13 0 1 0 14 / 13 0 0 1- 1 / 13 So we see that anything the subspace spanned by the α i are of the form ( a,b,c, 1 / 13(- 3 a + 14 b- c )). So we proceed to check α , β and γ . (4 ,- 5 , 9 , 1 / 13(- 12- 70- 9)) = (4 ,- 5 , 9 ,- 7) = α . (3 , 1 ,- 4 , 1 / 13(- 9 + 14 + 4)) = (3 , 1 ,- 4 , 9 / 13) 6 = β . (- 1 , 1 , , 1 / 13(3 + 14)) = (- 1 , 1 , , 17 / 13) 6 = γ . So α is the only vector in the space generated by the α i . (a) (b) Which of the vectors α , β , γ are in the subspace of C 4 spanned by the α i ? For the exact same reason, α is the only vector in the space generated by the α i . (b) (c) Does this suggest a theorem? Yes. Let F and K be fields such that F ⊂ K . Let { α 1 ,...,α m } be a set of vectors all of whose components are in F . Let v be any vector whose components are all elements of F then: v is in the F-vector space generated by the α i if and only if v is in the K-vector space generated by the α i . (c) Page 1 of 7 Math 235: Linear Algebra HW 5 Problem 2.6.3 Problem 2.6.3 Consider the vectors in R 4 defined by α 1 = (- 1 , , 1 , 2) , α 2 = (3 , 4 ,- 2 , 5) , α 3 = (1 , 4 , , 9) . Find a system of homogeneous linear equations for which the space of solutions is exactly the subspace of R 4 spanned by the three given vectors. First: row reduce the matrix with the α i as its rows: - 1 0 1 2 3 4- 2 5 1 4 9 → 1 0- 1- 2 0 1 1 / 4 11 / 4 0 0 So the system of equations we get is x 1- x 3- 2 x 4 = 0 x 2 + 1 / 4 x 3 + 11 / 4 x 4 = 0 Problem 2.6.5 Give an explicit description of the type (2-25) for the vectors β = ( b 1 ,b 2 ,b 3 ,b 4 ,b 5 ) ∈ R 5 which are a linear combination of the vectors α 1 = (1 , , 2 , 1 ,- 1) , α 2 = (- 1 , 2 ,- 4 , 2 , 0) , α 3 = (2 ,- 1 , 5 , 2 , 1) , α 4 = (2 , 1 , 3 , 5 , 2) ....
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235HW5 - Math 235 Linear Algebra HW 5 Problem 2.6.2 Problem...

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