MAT
201_2016_2_b

# N 1 1 t 1 1 t o select from the following 1 only d 2

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n 1 1 T , 1 - 1 T o Select from the following: 1. Only D. 2. Only C. 3. Only B. 4. Only A. 5. None of the above. Answer: 2 The null space is given by a b c R 3 : 0 1 - 1 0 - 1 1 a b c = 0 0 = a b c R 3 : b - c b - c = 0 0 = a b c R 3 : b = c = a b b R 3 = a 1 0 0 + b 0 1 1 : a, b R = span 1 0 0 , 0 1 1 . 7

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Since 1 0 0 , 0 1 1 is a linearly independent set, we have a basis for the null space. Thus every basis must have two elements which are column vectors in R 3 . We need only consider B and C. Since 1 0 0 / span n 0 1 - 1 T , 0 - 1 1 T o B does not span the null space and is not a basis for the null space. The set in C is linearly independent and span n 1 1 1 T , 0 1 1 T o = a a + b a + b R 3 : a, b R = α β β R 3 : α, β R by relabeling α = a and β = a + b (note that β can take on any real value independently of α ). Thus C provides a basis for the null space.
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