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**Unformatted text preview: **115A HW 3 SELECTED SOLUTIONS Sec 1.5. 9. Let u and v be distinct vectors in a vector space V . Show that { u, v } is linearly dependent if and only if u or v is a multiple of the other. If u and v are linearly dependent, then by definition there exist scalars a and b , not both zero, such that au + bv = 0. If a = 0, then bv = 0, so v = 0, and thus v = 0 u . If a 6 = 0, then u =- a- 1 bv . In either case, u and v are multiples of each other. Conversely, suppose that u is a multiple of v . Then u = cv for some constant v , so u- cv = 0. This shows that { u, v } is linearly dependent. 19. Observe that the problem is equivalent to showing that if { A 1 , . . . , A n } is linearly dependent then { A t 1 , . . . , A t n } is linearly dependent . If { A 1 , . . . , A n } is linearly dependent, then there exist scalars c i ∈ F not all zero such that c 1 A 1 + ··· + c n A n = 0 , where the 0 on the right-hand side of course denotes the zero matrix in M n × n ( F )....

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