Math 2250-1 Mon Oct 1 4.1 - 4.3 Concepts related to linear combinations of vectors.Recall our discussion from Friday. Can you remember the new terminology, which you want to become familiar with?A linear combination of the 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 is The span of 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 is\302\267 Recall that for vectors in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSJtRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 all linear combination questions can be reduced to matrix questions because any linear combination like the one on the left is actually just the matrix product on the right: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 .important: The vectors in the linear combination expression on the left are precisely the ordered columns of the matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, and the linear combination coefficients are the ordered entries in the vector LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2KFEiY0YnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiw2I1EhRicvRjdRJ25vcm1hbEYnToday's additional concept is linear independence/dependence for collections of vectors:Definition: a) 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 are linearly independent if and only if ("iff") the only way LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2J1EiMEYnLyUlYm9sZEdRJXRydWVGJy8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGNi9GNVEnbm9ybWFsRic= can be expressed as a linear combination of these vectors,LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYzLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictRjA2JS1GLDYoUSJ2RicvJSVib2xkR0Y3RjUvJSp1bmRlcmxpbmVHRjcvRjlRLGJvbGQtaXRhbGljRicvJStmb250d2VpZ2h0R1ElYm9sZEYnRjtGQy1JI21vR0YkNi1RIitGJ0ZBLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZaLyUpc3RyZXRjaHlHRlovJSpzeW1tZXRyaWNHRlovJShsYXJnZW9wR0ZaLyUubW92YWJsZWxpbWl0c0dGWi8lJ2FjY2VudEdGWi8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRmNvLUYwNiVGMi1GIzYkLUY+NiRRIjJGJ0ZBRkFGQy1GMDYlRkhGaG9GQ0ZULUZVNi1RIn5GJ0ZBRlhGZW5GZ25GaW5GW29GXW9GX28vRmJvUSYwLjBlbUYnL0Zlb0ZjcC1GVTYtUSMuLkYnRkFGWEZlbkZnbkZpbkZbb0Zdb0Zfb0Zhb0ZkcC1GVTYtUSIuRidGQUZYRmVuRmduRmluRltvRl1vRl9vRmJwRmRwRl9wRlQtRjA2JUYyLUYjNiQtRiw2JVEibkYnRjVGOEZBRkMtRjA2JUZIRl1xRkMtRlU2LVEiPUYnRkFGWEZlbkZnbkZpbkZbb0Zdb0Zfby9GYm9RLDAuMjc3Nzc3OGVtRicvRmVvRmhxLUY+NidGRUZLRk0vRjlGU0ZRRkE= ,is for 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 .
b) 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 are linearly dependent if there is some way to write LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2J1EiMEYnLyUlYm9sZEdRJXRydWVGJy8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGNi9GNVEnbm9ybWFsRic= as a linear combination of these 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where not all of the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImpGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSI9RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZILyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlctSSNtbkdGJDYkRj9GREZE . (We call such an equation a linear dependency. Note that if we have any such linear dependency, then any LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2KFEidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lKnVuZGVybGluZUdGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2JS1GLzYlUSJqRidGNS9GOlEnaXRhbGljRidGNUZELyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjpRJ25vcm1hbEYn with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImpGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSUmbmU7RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZILyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlctSSNtbkdGJDYkRj9GREZE is actually in the span of the remaining LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2KFEidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lKnVuZGVybGluZUdGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2JS1GLzYlUSJrRidGNS9GOlEnaXRhbGljRidGNUZELyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjpRJ25vcm1hbEYn with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RJSZuZTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJqRidGL0YyRjk= . We say that such a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2KFEidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lKnVuZGVybGluZUdGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2JS1GLzYlUSJqRidGNS9GOlEnaXRhbGljRidGNUZELyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjpRJ25vcm1hbEYn is linearly dependent on the remaining LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2KFEidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lKnVuZGVybGluZUdGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2JS1GLzYlUSJrRidGNS9GOlEnaXRhbGljRidGNUZELyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjpRJ25vcm1hbEYn .)Exercise 1) On Friday we showed that the vectors 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 and 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 span LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjJGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 (i.e. their span is all of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjJGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1). Recall how we checked this algebraically. Are these two vectors linearly independent? Exercise 2) Are the vectors 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 , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2KFEidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lKnVuZGVybGluZUdGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2JS1JI21uR0YkNiRRIjJGJy9GOlEnbm9ybWFsRidGNS9GOlEnaXRhbGljRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIj1GJ0ZFLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZSLyUpc3RyZXRjaHlHRlIvJSpzeW1tZXRyaWNHRlIvJShsYXJnZW9wR0ZSLyUubW92YWJsZWxpbWl0c0dGUi8lJ2FjY2VudEdGUi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRltvLUkobWZlbmNlZEdGJDYoLUYjNictSSdtdGFibGVHRiQ2Ni1JJG10ckdGJDYmLUkkbXRkR0YkNigtRkI2JFEiMUYnRkUvJSlyb3dhbGlnbkdRIUYnLyUsY29sdW1uYWxpZ25HRmFwLyUrZ3JvdXBhbGlnbkdGYXAvJShyb3dzcGFuR0ZecC8lK2NvbHVtbnNwYW5HRl5wRl9wRmJwRmRwLUZnbzYmLUZqbzYoLUZCNiRRIjNGJ0ZFRl9wRmJwRmRwRmZwRmhwRl9wRmJwRmRwLyUmYWxpZ25HUSVheGlzRicvRmBwUSliYXNlbGluZUYnL0ZjcFEmcmlnaHRGJy9GZXBRJ3xmcmxlZnR8aHJGJy8lL2FsaWdubWVudHNjb3BlR0Y0LyUsY29sdW1ud2lkdGhHUSVhdXRvRicvJSZ3aWR0aEdGXnIvJStyb3dzcGFjaW5nR1EmMS4wZXhGJy8lLmNvbHVtbnNwYWNpbmdHUSYwLjhlbUYnLyUpcm93bGluZXNHUSVub25lRicvJSxjb2x1bW5saW5lc0dGaXIvJSZmcmFtZUdGaXIvJS1mcmFtZXNwYWNpbmdHUSwwLjRlbX4wLjVleEYnLyUqZXF1YWxyb3dzR0ZSLyUtZXF1YWxjb2x1bW5zR0ZSLyUtZGlzcGxheXN0eWxlR0ZSLyUlc2lkZUdGZ3EvJTBtaW5sYWJlbHNwYWNpbmdHRmZyLyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYnLyUpcmVhZG9ubHlHRjQvJTBmb250X3N0eWxlX25hbWVHUSoyRH5PdXRwdXRGJ0ZFRkUvSSttc2VtYW50aWNzR0YkUSpDb2xWZWN0b3JGJy8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZjdEZF, 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 linearly independent? Is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2KFEidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lKnVuZGVybGluZUdGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2JS1JI21uR0YkNiRRIjFGJy9GOlEnbm9ybWFsRidGNS9GOlEnaXRhbGljRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZF linearly dependent on 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 ? Exercise 3) For linearly independent vectors 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, every vector LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2KFEidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRicvRjdRJ25vcm1hbEYn in their span can be written as 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 uniquely, i.e. for exactly one choice of linear combination coefficients 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 . This is not true if vectors are dependent. Explain why these facts are true. (This is why independent vectors are nice to have....in fact, if vectors are independent we honor them by calling them a basis for their span.)Exercise 4) Use properties of reduced row echelon form matrices to answer the following questions:4a) Why must more than two vectors in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjJGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 always be linearly dependent? 4b) Why can fewer than two vectors (i.e. one vector) not span LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjJGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 ?4c) If 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 are any two vectors in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjJGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 what is the condition on the reduced row echelon form of the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RKCZ0aW1lcztGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkdGK0Yv matrix 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 that guarantees they're linearly independent? That guarantees they span LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjJGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 ? That guarantees they're a basis for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjJGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 ?4d) Why must more than LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEiM0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv vectors in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjNGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 always be linearly dependent?
4e) Why can fewer than 3 vectors never span LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjNGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 ?4f) If you are given LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEiM0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv vectors 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 in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjNGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1, what is the condition on the reduced row echelon form of the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW5HRiQ2JFEiM0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RKCZ0aW1lcztGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkdGK0Yv matrix 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 that guarantees they're linearly independent? That guarantees they span LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjNGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 ? That guarantees they're a basis of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjNGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 ?4g) Can you generalize these questions and answers to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSJtRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 ?On Friday we consideredLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2KFEidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lKnVuZGVybGluZUdGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2JS1JI21uR0YkNiRRIjFGJy9GOlEnbm9ybWFsRidGNS9GOlEnaXRhbGljRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIj1GJ0ZFLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZSLyUpc3RyZXRjaHlHRlIvJSpzeW1tZXRyaWNHRlIvJShsYXJnZW9wR0ZSLyUubW92YWJsZWxpbWl0c0dGUi8lJ2FjY2VudEdGUi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRltvLUkobWZlbmNlZEdGJDYoLUYjNictSSdtdGFibGVHRiQ2Ny1JJG10ckdGJDYmLUkkbXRkR0YkNigtRkI2JFEiMkYnRkUvJSlyb3dhbGlnbkdRIUYnLyUsY29sdW1uYWxpZ25HRmFwLyUrZ3JvdXBhbGlnbkdGYXAvJShyb3dzcGFuR0ZELyUrY29sdW1uc3BhbkdGREZfcEZicEZkcC1GZ282Ji1Gam82KEZBRl9wRmJwRmRwRmZwRmhwRl9wRmJwRmRwRmpwLyUmYWxpZ25HUSVheGlzRicvRmBwUSliYXNlbGluZUYnL0ZjcFEmcmlnaHRGJy9GZXBRJ3xmcmxlZnR8aHJGJy8lL2FsaWdubWVudHNjb3BlR0Y0LyUsY29sdW1ud2lkdGhHUSVhdXRvRicvJSZ3aWR0aEdGW3IvJStyb3dzcGFjaW5nR1EmMS4wZXhGJy8lLmNvbHVtbnNwYWNpbmdHUSYwLjhlbUYnLyUpcm93bGluZXNHUSVub25lRicvJSxjb2x1bW5saW5lc0dGZnIvJSZmcmFtZUdGZnIvJS1mcmFtZXNwYWNpbmdHUSwwLjRlbX4wLjVleEYnLyUqZXF1YWxyb3dzR0ZSLyUtZXF1YWxjb2x1bW5zR0ZSLyUtZGlzcGxheXN0eWxlR0ZSLyUlc2lkZUdGZHEvJTBtaW5sYWJlbHNwYWNpbmdHRmNyLyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYnLyUpcmVhZG9ubHlHRjQvJTBmb250X3N0eWxlX25hbWVHUSoyRH5PdXRwdXRGJ0ZFRkUvSSttc2VtYW50aWNzR0YkUSpDb2xWZWN0b3JGJy8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZgdEZF, 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, LUkobWFjdGlvbkc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXJvd0dGJDYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2KFEidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNy8lKnVuZGVybGluZUdGNy8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiw2JS1JI21uR0YkNiRRIjNGJy9GPVEnbm9ybWFsRidGOC9GPVEnaXRhbGljRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIj1GJ0ZILyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZVLyUpc3RyZXRjaHlHRlUvJSpzeW1tZXRyaWNHRlUvJShsYXJnZW9wR0ZVLyUubW92YWJsZWxpbWl0c0dGVS8lJ2FjY2VudEdGVS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRl5vLUYsNictRiw2Ki1JKG1mZW5jZWRHRiQ2KC1GLDYnLUknbXRhYmxlR0YkNjctSSRtdHJHRiQ2Ji1JJG10ZEdGJDYoLUZFNiRRIjRGJ0ZILyUpcm93YWxpZ25HUSFGJy8lLGNvbHVtbmFsaWduR0ZocC8lK2dyb3VwYWxpZ25HRmhwLyUocm93c3BhbkdRIjFGJy8lK2NvbHVtbnNwYW5HRl9xRmZwRmlwRltxLUZecDYmLUZhcDYoLUYsNigtRlA2LVEqJnVtaW51czA7RidGSEZTRlZGWEZaRmZuRmhuRmpuL0Zdb1EsMC4yMjIyMjIyZW1GJy9GYG9GXHItRkU2JEZfcUZILyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYnLyUpcmVhZG9ubHlHRjcvJTBmb250X3N0eWxlX25hbWVHUSoyRH5PdXRwdXRGJ0ZIRmZwRmlwRltxRl1xRmBxRmZwRmlwRltxLUZecDYmLUZhcDYoLUZFNiRRIzExRidGSEZmcEZpcEZbcUZdcUZgcUZmcEZpcEZbcS8lJmFsaWduR1ElYXhpc0YnL0ZncFEpYmFzZWxpbmVGJy9GanBRJnJpZ2h0RicvRlxxUSd8ZnJsZWZ0fGhyRicvJS9hbGlnbm1lbnRzY29wZUdGNy8lLGNvbHVtbndpZHRoR1ElYXV0b0YnLyUmd2lkdGhHRlx0LyUrcm93c3BhY2luZ0dRJjEuMGV4RicvJS5jb2x1bW5zcGFjaW5nR1EmMC44ZW1GJy8lKXJvd2xpbmVzR1Elbm9uZUYnLyUsY29sdW1ubGluZXNHRmd0LyUmZnJhbWVHRmd0LyUtZnJhbWVzcGFjaW5nR1EsMC40ZW1+MC41ZXhGJy8lKmVxdWFscm93c0dGVS8lLWVxdWFsY29sdW1uc0dGVS8lLWRpc3BsYXlzdHlsZUdGVS8lJXNpZGVHRmVzLyUwbWlubGFiZWxzcGFjaW5nR0ZkdEZgckZjckZlckZIRkgvSSttc2VtYW50aWNzR0YkUSpDb2xWZWN0b3JGJy8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZpdS1GUDYtUSIsRidGSEZTL0ZXRjdGWEZaRmZuRmhuRmpuL0Zdb1EmMC4wZW1GJy9GYG9RLDAuMzMzMzMzM2VtRictRlA2LVEifkYnRkhGU0ZWRlhGWkZmbkZobkZqbkZmdi9GYG9GZ3YtRjI2KFEiYkYnRjVGOEY6RjxGP0ZPLUZQNi9GXHdGNS9GPUZBRj9GU0ZWRlhGWkZmbkZobkZqbkZmdkZdd0ZIRml1LUYsNiYtRjI2I0ZocC1GLDYmRmZ3LUYsNiQtRmZvNigtRiw2JkZmdy1GW3A2Ny1GXnA2Ji1GYXA2KC1GLDYlLUZFNiRRIjZGJ0ZIRjhGSkZmcEZpcEZbcUZdcUZgcUZmcEZpcEZbcS1GXnA2Ji1GYXA2KC1GLDYkRkRGSEZmcEZpcEZbcUZdcUZgcUZmcEZpcEZbcS1GXnA2Ji1GYXA2KEZCRmZwRmlwRltxRl1xRmBxRmZwRmlwRltxRl9zRmJzRmRzRmZzRmhzRmpzRl10Rl90RmJ0RmV0Rmh0Rmp0Rlx1Rl91RmF1RmN1RmV1Rmd1RmZ3RkhGSEZpdUZcdkZfdkZpdUZIRmZ3RkhGZndGSC1GUDYtUSomRWxlbWVudDtGJ0ZIRlNGVkZYRlpGZm5GaG5Gam5GXG9GX29GSEZpdS1JJW1zdXBHRiQ2JS1GMjYlUSgmIzg0Nzc7RicvRjlGVUZIRkIvJTFzdXBlcnNjcmlwdHNoaWZ0R0ZORkgvJSthY3Rpb250eXBlR1EucnRhYmxlYWRkcmVzc0YnBy Exercise 4 and before doing any work, we know that 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 must be linearly dependent. In fact, on Friday we found some explicit dependencies between these four vectors. We'll take a slightly different look at those questions today. All we need is the reduced row echelon form 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 linear dependency 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 between these four vectors corresponds to a solution of a homogeneous matrix equation, with the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW5HRiQ2JFEiM0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RKCZ0aW1lcztGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkctRiw2JFEiNEYnRi9GLw== matrix on the left. To find these, we would augment that matrix with a column of zeroes and reduce - yielding the matrix on the right augmented by a column of zeroes. By the same reasoning, those final homogeneous solutions correspond to column dependencies for the right side matrix. Upshot: Column dependencies (and independencies) for matrices that differ by elementary row operations are identical! To emphasize this point, we will explicitly make the augmented matrices (this time) even though we could do the math without doing so: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Exercise 5)5a) Use the reasoning above to write LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2KFEiYkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRicvRjdRJ25vcm1hbEYn as a linear combination of 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 .5b) Write LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2KFEidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lKnVuZGVybGluZUdGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2JS1JI21uR0YkNiRRIjNGJy9GOlEnbm9ybWFsRidGNS9GOlEnaXRhbGljRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZF as a linear combination of 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 .5c) In how many ways can you cull the original four vectors LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUklbXN1YkdGJDYlLUkjbWlHRiQ2KFEidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lKnVuZGVybGluZUdGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2JS1JI21uR0YkNiRRIjFGJy9GOlEnbm9ybWFsRidGNS9GOlEnaXRhbGljRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIixGJ0ZFLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRlIvJSpzeW1tZXRyaWNHRlIvJShsYXJnZW9wR0ZSLyUubW92YWJsZWxpbWl0c0dGUi8lJ2FjY2VudEdGUi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVGLi1GIzYlLUZCNiRRIjJGJ0ZFRjVGR0ZJRkwtRiw2JUYuLUYjNiUtRkI2JFEiM0YnRkVGNUZHRklGTC1GLzYoUSJiRidGMkY1RjdGOUY8RkU= down to just two independent vectors, so that the span of the final two is the same as the span of the original four? (These final two vectors will be a basis for the span.)Exercise 6) Explain why the span of 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 is not all of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjNGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 .Exercise 7a) The span of 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 is actually a plane through the origin, in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjNGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1. Pick any basis of two vectors from the original four, and find the implicit equation of this plane. Hint, if we use 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, and if we want to find out whether 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 is in the span, we should reduce the 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.7b) Verify that all four of the original vectors lie on your plane!JSFHTTdSMApJN1JUQUJMRV9TQVZFLzQzMDcwNTEwMDhYKiUpYW55dGhpbmdHNiJGJVtnbCEjJSEhISIkIiQmJSJtRzYkIiIiRikmRic2JCIiI0YpJkYnNiQiIiRGKUYlTTdSMApJN1JUQUJMRV9TQVZFLzQzMzM4NjEyMzJYKiUpYW55dGhpbmdHNiJGJVtnbCEjJSEhISIkIiQmJSJtRzYkIiIiRikmRic2JCIiI0YpJkYnNiQiIiRGKUYl