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Quiz 3

# Quiz 3 - Math 3A(44351 Quiz 3 Problem 1 Let A1 = Show that...

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Math 3A (44351) Quiz 3 Problem 1. Let A 1 = 1 0 3 4 , A 2 = 4 0 - 2 3 , A 3 = 0 0 1 2 Show that T = 7 0 0 1 Span( A 1 , A 2 , A 3 ) . Do { A 1 , A 2 , A 3 } span all of R 2 x 2 ? Explain. Solution: 3 A 1 + A 2 - 7 A 3 = 3 0 9 12 + 4 0 - 2 3 - 0 0 7 14 = 7 0 0 1 and so T is a linear combination of A 1 , A 2 , A 3 , and thus T Span( A 1 , A 2 , A 3 ). Since all three vectors have a top-right component of 0, it is clear that the vector 0 1 0 0 , which is in R 2 x 2 , is not in Span( A 1 , A 2 , A 3 ), so no, they do not span all of R 2 x 2 Problem 2. Suppose that { v 1 , v 2 , . . . , v k } , with k > 2, is a set of lin- early independent vectors in R n . Prove that the set { v 2 , . . . , v k } must also be linearly independent. Solution: Suppose that c 2 v 2 + . . . + c k v k = 0 for some set of constants
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