102_solutions4

# 102_solutions4 - MATH 102 SOLUTIONS TO HW#4 Section 3.1...

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MATH 102 SOLUTIONS TO HW #4 Section 3.1, problem 6. First we find a basis for the space of vectors orthogonal to both a = (1 , 1 , 1) and b = (1 , - 1 , 0). this space is nothing other that the null-space of the matrix: A = 1 1 1 1 - 1 0 . To compute it, after the following Gaussian elimination steps: (1) - 1 × { Row 1 } + { Row 2 } ⇒ { Row 2 } (2) 1 2 × { Row 2 } + { Row 1 } ⇒ { Row 1 } (3) - 1 2 × { Row 2 } ⇒ { Row 2 } we are left with the reduces matrix: R = 1 0 1 2 0 1 1 2 . From this, a basis for the null-space of A is easily computed to be: c = - 1 2 - 1 2 1 . To normalize these vectors, we just divide through by their lengths. Doing this yields the orthonormal set: ˆ a = 1 3 1 3 1 3 , ˆ b = 1 2 - 1 2 0 , ˆ c = - 1 6 - 1 6 2 6 . Section 3.1, problem 12. This problem again asks to compute the null-space of A . Recall that N ( A ) = [ R ( A )] . After performing the reduction step: (1) - 1 × { Row 1 } + { Row 2 } ⇒ { Row 2 } we have the reduced matrix: R = 1 0 2 0 1 2 .

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