How do I show only using result about system of homogeneous linear equation (i mean only using calculus technique and not theorems about vectorial spaces) that 4 vectors are linear dependent using 3 generators? my teacher said in this case i have 3 equation with 4 variables but i can't see this . Why 4 variables and not 3 (generators)?
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You might want to state the "result about a system of homogeneous linear equations" to make it more clear. – mmcrjx Feb 08 '21 at 14:32
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only using linear combinations and calculus , no theorems about vectorial spaces – Alfredo Cozzolini Feb 08 '21 at 14:55
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1Yes add that result to your question. – mmcrjx Feb 08 '21 at 14:58
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Sorry but I'm not understanding "4 vectors are linear dependent using 3 generators" – Vajra Feb 08 '21 at 15:01
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there are 3 vector that generate my entire space and 4 vectors. and i have to show that these 4 vector are linear depent – Alfredo Cozzolini Feb 08 '21 at 15:04
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If we assume that the three vectors that generate the space and a vector $u\in V$ are linealy independent the matrix $M=(v_1|v_2|v_3|u)$ would have rank $4$ but $M\in \operatorname{M}(3\times 4,\mathbb K)$ so... – Vajra Feb 08 '21 at 15:14
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so the max rank would be 3 and that is impossible right? – Alfredo Cozzolini Feb 08 '21 at 15:42
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Seems like a duplicate of this question. – mmcrjx Feb 08 '21 at 22:33
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oh sorry i didn't know about rhe existence of this question . Anyway thanks for having shared this with me there are a lot of interesting proofs – Alfredo Cozzolini Feb 09 '21 at 08:36