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# a6 - Explain your answer 6 Extend the pair of vectors a = 1...

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ASSIGNMENT 6 EE441 KUMAR Fall 2007 Due: Monday, Oct. 15 1. Find a basis each for the rowspace, columnspace and nullspace of the matrix A = 1 2 0 - 1 1 3 1 1 2 5 1 0 3 6 0 0 . 2. What is the coordinate representation of the vector [1 0 0] T with respect to the basis A = n [1 2 3] T , [0 1 2] T , [0 0 1] T o ? 3. Consider the following two basis for < 2 : A = (" 1 2 # , " 3 4 #) and B = (" 1 1 # , " 1 2 #) Find a matrix P such that [ x ] B = P [ x ] A for all x ∈ < 2 . 4. What is the dimension of the vector space of all 3 × 3 upper triangular matrices ? (A is upper triangular if and only if a 21 = a 31 = a 32 = 0). 5. If the n columns of an n × n matrix A are linearly dependent, the same will always be true of the n columns of the matrix A 2 . (i) True (ii) False Explain your answer.

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Unformatted text preview: Explain your answer. 6. Extend the pair of vectors a = 1 2 3 , b = 2 3 7 to a basis for < 3 , i.e., ﬁnd a vector c such that { a ,b ,c } are a set of linearly independent vectors. 1 7. Find a basis for the vector space W = < a 1 , a 2 , a 3 , a 4 > spanned by the (4 × 1) vectors a 1 = 1 3-1-2 , a 2 = 2 6-1-3 , a 3 = 3 8-3-5 and a 4 = -1-2 1 1 . 2...
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