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SolutionQuiz3.pdf - SolutionQuiz3 October 26 2020 0.1...

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SolutionQuiz3 October 26, 2020 0.1Solution of Quiz 3 [1]:importnumpyasnp importsympyassy importscipy.linalgasla importmatplotlib.pyplotasplt 0.1.1Exercise 1 a) Letuandvbe vectors inRn . Ifuhas only negative entries, thenu·v≤0.
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b) Let u and v be vectors in R n . If v has only positive entries, thenu·v≥0.
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c) Letuandvbe vectors inRn . If u and v have only non-negative entries, thenu·v≥0.
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d) Letuandvbe vectors inRn. Ifuandvhave only negative entries, thenu·v <0.
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0.1.2Exercise 2 a) Which of the following pairs of vectors are a basis for the nullspace N(A) of the matrix
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[2]: 10−4−3 −21135 015−1 [3]:A.nullspace() [3]:[Matrix([ [ 4], [-5], [ 1], [ 0]]), Matrix([ [3], [1], [0], [1]])] For the other b),c) and d) a check can be preformed similarly b) Which of the following pairs of vectors are a basis for the row space R(AT ) of the matrix
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c) Which of the following pairs of vectors are a basis for the nullspace N( A )of the matrix
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d) Which of the following pairs of vectors are a basis for the row space R(AT ) of the matrix
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2
Answer:10−1 3 7 3 , 0 −3 1 2 Any scalar multiples of these vectors is also correct. 0.1.3Exercise 3
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