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

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Unformatted text preview: Math 3A (44351) Quiz 3 Problem 1. Let A1 = Show that T = all of R2x2 ? Explain. 1 0 3 4 7 0 0 1 , A2 = 4 0 -2 3 , A3 = 0 0 1 2 Span(A1 , A2 , A3 ). Do {A1 , A2 , A3 } span Solution: 3A1 + A2 - 7A3 = 3 0 9 12 + 4 0 -2 3 - 0 0 7 14 = 7 0 0 1 and so T is a linear combination of A1 , A2 , A3 , and thus T Span(A1 , A2 , A3 ). Since all three vectors have a top-right component of 0, it is clear 0 1 , which is in R2x2 , is not in Span(A1 , A2 , A3 ), that the vector 0 0 so no, they do not span all of R2x2 Problem 2. Suppose that {v1 , v2 , . . . , vk }, with k > 2, is a set of linearly independent vectors in Rn . Prove that the set {v2 , . . . , vk } must also be linearly independent. Solution: Suppose that c2 v2 +. . . +ck vk = 0 for some set of constants c2 , . . . , ck . Then, if we set c1 = 0, it is clear that c1 v1 + c2 v2 + . . . + ck vk = 0. Since {v1 , v2 , . . . , vk } is linearly independent, and this linear combination is equal to 0, it must be the trivial combination, that is, every coefficient must be 0, so c1 = c2 = ... = ck = 0. Thus we have shown that c2 v2 + . . . + ck vk = 0 implies that c2 = ... = ck = 0, and so, {v2 , . . . , vk } is also linearly independent. 1 ...
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This note was uploaded on 05/08/2010 for the course MAT MATH taught by Professor Pantano during the Spring '10 term at UC Irvine.

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