Maticový zápis soustavy lineárních rovnic

V praktických aplikacích většinou volíme maticový zápis osustavy rovnic. Je přehlednější a je snadnější jej převést do syntaxe používaného software (Matlab, Maple,...). Rozepsání do složek vektoru neznámých provedeme roznásobením. Je třeba mít na paměti, že násobení matic není komutativní. Je rozdíl, zda je vektor neznámých x napsán zleva nebo zprava. Obvykle je volen zápis [color=#0000ff][i]Ax = b[/i][/color] a pokud není řečeno jinak, software toto pořadí činitelů předpokládá. Pokud je zadání ve tvaru součinu [color=#0000ff][i]xA[/i][/color], rovnici transponujeme. Součin transponujeme transponováním každého členu a záměnou pořadí. A právě o to nám jde. [br][br][color=#073763][size=150]Příklad:[/size][/color][br]Určete všechna řešení homogenní soustavy lineárních rovnic[br][img]data:image/png;base64,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[/img][br][color=#073763][size=150]Řešení:[/size][/color] Vynásobíme-li matici [i]A[/i] vektorem neznámých [i]x[/i], dostaneme na levé straně rovnice vektor. Dva vektory se sobě rovnají, rovnají-li se všechny jejich složky. Pro vektory z R[sup]2[/sup] máme tedy dvě lineární rovnice pro dvě neznámé [i]x[/i][sub]1[/sub], [i]x[/i][sub]2[/sub].[br][img]data:image/png;base64,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[/img][br]Hodnost matice A je dva, protože řádky jsou lineárně závislé, odtud i rovnice jsou lineárně závislé a ve skutečnosti máme jej jednu lineární podmínku pro uspořídanou dvojici [i]x[/i][sub]1[/sub], [i]x[/i][sub]2[/sub]. Řešení je nekonečně mnoho, závislých na jednom parametru. Zvolíme např. [i]x[/i][sub]1[/sub] jako reálný parametr [i]t[/i].[br][img]data:image/png;base64,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[/img]
Příklady k procvičení
Příklad 1
Postup řešení
Všechny příklady jsou tak jednoduché, že ani není třeba používat nástroje GeoGebry [code]SchodovityTvar[/code] pro určení Gaussovy eliminace. [br][br][size=150][color=#0B5394]Příklad 1 (bez software):[/color][/size][br][img]data:image/png;base64,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[/img][br]Řešení je patrné ihned, z roznásobení [i]x.Q[/i]. Pozor, nyní násobíme vektorem neznámých zleva, proto musí být zapsán v řádku, jinak by výsledný součin neexistoval.[br][img]data:image/png;base64,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[/img][br]Hodnost matice je jedna, tj. dostáváme jej jednu rovnici pro dvě neznámé a řešení tvoří jednoparametrický systém [br][img]data:image/png;base64,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[/img].[br][size=150][color=#0B5394]Příklad 1(s GeoGebrou): [/color][/size][br]Stejný výsledek dostaneme i ze schodovité matice [color=#0B5394]QTel[/color] (applet výše). Nejprve matici [color=#0B5394][i]Q [/i][/color]transponujeme, abychom vyměnili pořadí činitelů. Příkazem [code]SchodovityTvar[/code](QT) získáme ekvivalentní matici [color=#0B5394]QTel[/color], z níž přepisem do soustavy rovnic [br][img]data:image/png;base64,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[/img][br]vidíme výsledek [i]x[/i][sub]1[/sub]=0, tj. [i]x[/i][sub]2[/sub] může být jakékoliv reálné číslo.
Řešení
Hašek, R.: Lineární algebra, kapitola "Soustavy lineárních rovnic"

Information: Maticový zápis soustavy lineárních rovnic