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let {v1-v2} be linearly independent in a vectorspace V- Show that if a.docx

let {v1,v2} be linearly independent in a vectorspace V. Show that if a vector v3 is not of the form av1+av2, then the set {v1,v2,v3} is linearly independent
Solution
Say, to the contrary that the set {v1,v2,v3} is linearly dependent, then there exists a non-trivial solution to the equation a1v1+a2v2+a3v3=0. Say a3=0 then the equation a1v1+a2v2=0 has a non-trivial solution, which implies that v1 and v2 are linearly dependent, which is absurd. Thus, a3 is not 0. Hence, v3=(-a1/a3)v1+(-a1/a2)v2, which means that v3 is a linear combination of v1 and v2, which is a contradiction. Therefore, the set {v1,v2,v3} is linearly independent.
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let {v1,v2} be linearly independent in a vectorspace V. Show that if a vector v3 is not of the form av1+av2, then the set {v1,v2,v3} is linearly independent
Solution
Say, to the contrary that the set {v1,v2,v3} is linearly dependent, then there exists a non-trivial solution to the equation a1v1+a2v2+a3v3=0. Say a3=0 then the equation a1v1+a2v2=0 has a non-trivial solution, which implies that v1 and v2 are linearly dependent, which is absurd. Thus, a3 is not 0. Hence, v3=(-a1/a3)v1+(-a1/a2)v2, which means that v3 is a linear combination of v1 and v2, which is a contradiction. Therefore, the set {v1,v2,v3} is linearly independent.
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