QBUS1040 Tutorial 5 Semester 2, 2022
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
QBUS1040
Tutorial 5
Semester 2, 2022
Exercise 1: (Norm of a linear combination of orthonormal vectors)
Suppose a1 , . . . ,ak are orthonormal n-vectors, and x = β 1 a1 + ··· +βk ak where β1 , . . . ,βk are scalars. Express ∥x∥ in terms of β = (β1 , . . . ,βk ).
Exercise 2: (Early termination of Gram-Schmidt algorithm)
When the Gram-Schmidt algorithm is run on a particular list of ten 15-vectors, it terminates at iteration 5 (and hence q˜5 = 0). Which of the following must be true?
(a) a2 ,a3 ,a4 are linearly independent.
(b) a1 ,a2 ,a5 are linearly dependent.
(c) a1 ,a2 ,a3 ,a4 ,a5 are linearly dependent.
(d) a4 is nonzero.
Exercise 3: (Gram-Schmidt algorithm: Code it up!)
Implement the Gram-Schmidt algorithm using Python. You may want to define some functions in the algorithm.
Exercise 4: (Linear independence of stacked vectors.)
Consider the stacked vectors
c1 = [b(a)1(1)] , . . . , ck = [b(a)k(k)] ,
where a1 , . . . ,ak are n-vectors and b1 , . . . ,bk are m-vectors.
(a) Suppose a1 , . . . ,ak are linearly independent. (We make no assumptions about the vectors b1 , . . . ,bk .) Can we conclude that the stacked vectors c1 , . . . ,ck are linearly independent?
(b) Now suppose that a1 , . . . ,ak are linearly dependent, (Again, with no assumptions about b1 , . . . ,bk .) Can we conclude that the stacked vectors c1 , . . . ,ck are linearly dependent?
2022-09-20