MATH 330 H45

# MATH 330 H45 - Math 330 Homework 4.5(Pages 261,262(6 Find a...

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Unformatted text preview: Math 330 Homework 4.5 (Pages 261,262) (6) Find a basis and state the dimension of the following: 3 a + 6 b- c 6 a- 2 b- 2 c- 9 a + 5 b + 3 c- 3 a + b + c : a, b, c ∈ R SOLUTION: 3 a + 6 b- c 6 a- 2 b- 2 c- 9 a + 5 b + 3 c- 3 a + b + c = a 3 6- 9- 3 + b 6- 2 5 1 c - 1- 2 3 1 Shows that the three indicated column vectors have a span which produces the subspace. However, unless these three vectors are linearly independent, they do not form a basis: 3 6- 1 6- 2- 2- 9 5 3- 3 1 1 ( R 2 ,R 3 ,R 4) ← ( R 2- 2 R 1 ,R 3+3 R 1 ,R 4+ R 1) = ⇒ 3 6- 1- 14 23 7 This shows we do not have three pivots so the vectors are dependent. However, since the first two vectors are not multiples of each other they are independent and hence our space has dimension 2 with basis 3 6- 9- 3 , 6- 2 5 1 (8) Find a basis and state the dimension of...
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MATH 330 H45 - Math 330 Homework 4.5(Pages 261,262(6 Find a...

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