Аналитическая геометрия и линейная алгебра. I семестр. Курс лекций. Кирсанов А.А. - 3 стр.

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ÑÎÄÅÐÆÀÍÈÅ
1. Ìàòðèöû
1.1. Îïðåäåëåíèå è âèäû ìàòðèö ..................................................... 7
1.2. Òðàíñïîíèðîâàííàÿ ìàòðèöà ..................................................10
1.3. Ñëîæåíèå ìàòðèö. Óìíîæåíèå ìàòðèöû íà ÷èñëî. ................11
1.4. Ëèíåéíàÿ çàâèñèìîñòü ìàòðèö ................................................13
1.5. Ñèìâîë
Σ
. Ïðàâèëî ñóììèðîâàíèÿ Ýéíøòåéíà ....................17
1.6. Óìíîæåíèå ìàòðèö ...................................................................20
1.7. Ýëåìåíòàðíûå ïðåîáðàçîâàíèÿ. Ýëåìåíòàðíûå ìàòðèöû. ....23
1.8. Âûðîæäåííûå è íåâûðîæäåííûå ìàòðèöû. ............................26
1.9. Îáðàòíàÿ ìàòðèöà. ...................................................................29
1.10. Ðàíã ìàòðèöû ..........................................................................32
1.11. Îñíîâíûå òåîðåìû î ðàíãå ìàòðèöû .....................................35
2. Îïðåäåëèòåëè
2.1. Îïðåäåëèòåëè II è III ïîðÿäêîâ ...............................................39
2.2. Îïðåäåëèòåëü ìàòðèöû
n
-ãî ïîðÿäêà ....................................41
2.3. Ñâîéñòâà îïðåäåëèòåëåé ..........................................................46
2.4. Âû÷èñëåíèå îáðàòíîé ìàòðèöû ñ ïîìîùüþ îïðåäåëèòåëÿ ...54
3. Ñèñòåìû ëèíåéíûõ óðàâíåíèé
3.1. Îïðåäåëåíèå è âèäû ñèñòåì ëèíåéíûõ óðàâíåíèé..................55
3.2. Ñèñòåìû ëèíåéíûõ óðàâíåíèé ñ
nm =
...................................59
3.4. Ïðàâèëî Êðàìåðà .....................................................................60
3.5. Òåîðåìà Êðîíåêåðà-Êàïåëëè ...................................................63
3.6. Îáùåå ïðàâèëî íàõîæäåíèÿ ðåøåíèé .....................................64
3.7. Ïðèâåä¸ííàÿ ñèñòåìà ëèíåéíûõ óðàâíåíèé ............................66
3.8. Îáùåå ðåøåíèå ñèñòåìû ëèíåéíûõ óðàâíåíèé .......................71
4. Âåêòîðíàÿ àëãåáðà
4.1. Îïðåäåëåíèå âåêòîðà è ëèíåéíûå îïåðàöèè íàä âåêòîðàìè .75
4.2. Ëèíåéíàÿ çàâèñèìîñòü âåêòîðîâ. ............................................ 79
4.3. Áàçèñ. .........................................................................................81
4.4. Ñèñòåìû êîîðäèíàò ..................................................................82
4.4.1. Äåêàðòîâà (àôôèííàÿ) ñèñòåìà êîîðäèíàò .......................82
4.4.2. Äåëåíèå îòðåçêà â äàííîì îòíîøåíèè. ............................. 84
4.4.3. Äåêàðòîâà ïðÿìîóãîëüíàÿ ñèñòåìà êîîðäèíàò. ............... 85
4.4.4. Ïîëÿðíàÿ ñèñòåìà êîîðäèíàò. ...........................................85
4.4.5. Öèëèíäðè÷åñêàÿ è ñôåðè÷åñêàÿ ñèñòåìû êîîðäèíàò. ......87
4.5. Ïðåîáðàçîâàíèå êîîðäèíàò ..................................................... 88
4.5.1. Ïàðàëëåëüíûé ïåðåíîñ ÏÑÊ íà ïëîñêîñòè. .....................88
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                                       ÑÎÄÅÐÆÀÍÈÅ

1. Ìàòðèöû
    1.1. Îïðåäåëåíèå è âèäû ìàòðèö ..................................................... 7
    1.2. Òðàíñïîíèðîâàííàÿ ìàòðèöà ..................................................10
    1.3. Ñëîæåíèå ìàòðèö. Óìíîæåíèå ìàòðèöû íà ÷èñëî. ................11
    1.4. Ëèíåéíàÿ çàâèñèìîñòü ìàòðèö ................................................13
    1.5. Ñèìâîë Σ . Ïðàâèëî ñóììèðîâàíèÿ Ýéíøòåéíà ....................17
    1.6. Óìíîæåíèå ìàòðèö ...................................................................20
    1.7. Ýëåìåíòàðíûå ïðåîáðàçîâàíèÿ. Ýëåìåíòàðíûå ìàòðèöû. ....23
    1.8. Âûðîæäåííûå è íåâûðîæäåííûå ìàòðèöû. ............................26
    1.9. Îáðàòíàÿ ìàòðèöà. ...................................................................29
    1.10. Ðàíã ìàòðèöû ..........................................................................32
    1.11. Îñíîâíûå òåîðåìû î ðàíãå ìàòðèöû .....................................35
2. Îïðåäåëèòåëè
    2.1. Îïðåäåëèòåëè II è III ïîðÿäêîâ ...............................................39
    2.2. Îïðåäåëèòåëü ìàòðèöû n -ãî ïîðÿäêà ....................................41
    2.3. Ñâîéñòâà îïðåäåëèòåëåé ..........................................................46
    2.4. Âû÷èñëåíèå îáðàòíîé ìàòðèöû ñ ïîìîùüþ îïðåäåëèòåëÿ ...54
3. Ñèñòåìû ëèíåéíûõ óðàâíåíèé
    3.1. Îïðåäåëåíèå è âèäû ñèñòåì ëèíåéíûõ óðàâíåíèé ..................55
    3.2. Ñèñòåìû ëèíåéíûõ óðàâíåíèé ñ m = n ...................................59
    3.4. Ïðàâèëî Êðàìåðà .....................................................................60
    3.5. Òåîðåìà Êðîíåêåðà-Êàïåëëè ...................................................63
    3.6. Îáùåå ïðàâèëî íàõîæäåíèÿ ðåøåíèé .....................................64
    3.7. Ïðèâåä¸ííàÿ ñèñòåìà ëèíåéíûõ óðàâíåíèé ............................66
    3.8. Îáùåå ðåøåíèå ñèñòåìû ëèíåéíûõ óðàâíåíèé .......................71
4. Âåêòîðíàÿ àëãåáðà
    4.1. Îïðåäåëåíèå âåêòîðà è ëèíåéíûå îïåðàöèè íàä âåêòîðàìè .75
    4.2. Ëèíåéíàÿ çàâèñèìîñòü âåêòîðîâ. ............................................79
    4.3. Áàçèñ. .........................................................................................81
    4.4. Ñèñòåìû êîîðäèíàò ..................................................................82
      4.4.1. Äåêàðòîâà (àôôèííàÿ) ñèñòåìà êîîðäèíàò .......................82
      4.4.2. Äåëåíèå îòðåçêà â äàííîì îòíîøåíèè. .............................84
      4.4.3. Äåêàðòîâà ïðÿìîóãîëüíàÿ ñèñòåìà êîîðäèíàò. ...............85
      4.4.4. Ïîëÿðíàÿ ñèñòåìà êîîðäèíàò. ...........................................85
      4.4.5. Öèëèíäðè÷åñêàÿ è ñôåðè÷åñêàÿ ñèñòåìû êîîðäèíàò. ......87
    4.5. Ïðåîáðàçîâàíèå êîîðäèíàò .....................................................88
      4.5.1. Ïàðàëëåëüíûé ïåðåíîñ ÏÑÊ íà ïëîñêîñòè. .....................88

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