Math 2250-1 Wed Oct 3 4.2-4.4 vector spaces, subspaces, bases and dimensionExercise 1) Vocabulary review:A linear combination of the 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 is The span of 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 isThe vectors 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 are linearly independent iffThe vectors 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 are linearly dependent iffLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is a vector space meansLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is a subspace of a vector space LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= means\302\267 New concepts for today: Definition: The vectors LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYvLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2KFEidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNy8lKnVuZGVybGluZUdGNy8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2JC1JI21uR0YkNiRRIjFGJy9GPVEnbm9ybWFsRidGSC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiLEYnRkgvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGUy8lKnN5bW1ldHJpY0dGUy8lKGxhcmdlb3BHRlMvJS5tb3ZhYmxlbGltaXRzR0ZTLyUnYWNjZW50R0ZTLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRk42LVEifkYnRkhGUS9GVUZTRlZGWEZaRmZuRmhuRmpuL0Zeb0Zcby1GMDYlRjItRiM2JC1GRTYkUSIyRidGSEZIRkpGTUZgb0Zgby1GTjYtUSMuLkYnRkhGUUZjb0ZWRlhGWkZmbkZobi9GW29RLDAuMjIyMjIyMmVtRidGZG8tRk42LVEiLkYnRkhGUUZjb0ZWRlhGWkZmbkZobkZqbkZkb0Zgby1GMDYlRjItRiM2JC1GLDYlUSJuRidGOC9GPVEnaXRhbGljRidGSEZKRkg= are called a basis for a subspace LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= iff they span LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and are linearly independent. (We talked about a basis for the span of a collection of vectors before, which this definition generalizes.)Definition: The dimension of a subspace LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the number of vectors in a basis for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= . (It turns out that all bases for a subspace always have the same number of vectors.)Yesterday we showed in an example that the span of two particular vectors was a subspace. Exactly the same idea shows a much more general fact:Theorem 1) The span of any collection of vectors 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 in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSJtRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 (or actually the span of any 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 in some vector space LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=) is closed under addition and scalar multiplication, and so is a subspace of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSJtRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 (or LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIilGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdRJmZhbHNlRicvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EsMC4xNjY2NjY3ZW1GJy8lJ3JzcGFjZUdGTEY5. why: We need to show 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 is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtaUdGJDYlUScmIzk0NTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjdGNy1JI21vR0YkNi1RIn5GJ0Y3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTkY3closed under addition and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtaUdGJDYlUScmIzk0NjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjdGN0Y3 closed under scalar multiplication. Let 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtaUdGJDYlUScmIzk0NTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjdGNy1JI21vR0YkNi1RIn5GJ0Y3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTkY6Rjc=Then using the algebra properties of addition and scalar multiplication to rearrange and collect terms,LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY2LUkjbWlHRiQ2KFEidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSNtb0dGJDYtUSIrRicvRjdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlMtRiw2KFEid0YnRi9GMkY0RjZGOS1GPTYtUSI9RidGQEZCRkVGR0ZJRktGTUZPL0ZSUSwwLjI3Nzc3NzhlbUYnL0ZVRmduLUkobWZlbmNlZEdGJDYkLUYjNiYtSSVtc3ViR0YkNiUtRiw2JVEiY0YnRjIvRjdRJ2l0YWxpY0YnLUYjNiQtSSNtbkdGJDYkUSIxRidGQEZALyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPC1GX282JS1GLDYlUSJkRidGMkZkby1GIzYlRmhvRjJGZG9GXHBGQEZALUZfbzYlRitGZm9GXHBGPC1Gam42JC1GIzYmLUZfbzYlRmFvLUYjNiQtRmlvNiRRIjJGJ0ZARkBGXHBGPC1GX282JUZhcC1GIzYlRmBxRjJGZG9GXHBGQEZALUY9Ni1RIn5GJ0ZARkJGRUZHRklGS0ZNRk8vRlJRJjAuMGVtRicvRlVGW3ItRl9vNiVGK0ZecUZccEY8RmdxRmdxLUY9Ni1RIy4uRidGQEZCRkVGR0ZJRktGTUZPRlFGXHItRj02LVEiLkYnRkBGQkZFRkdGSUZLRk1GT0ZqcUZcckY8LUZqbjYkLUYjNiYtRl9vNiVGYW8tRiM2JC1GLDYlUSJuRidGMkZkb0ZARlxwRjwtRl9vNiVGYXAtRiM2JUZdc0YyRmRvRlxwRkBGQEZncS1GX282JUYrRltzRlxwRkA= .Thus LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2KFEidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSNtb0dGJDYtUSIrRicvRjdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlMtRiw2KFEid0YnRi9GMkY0RjZGOS1GLDYjUSFGJ0ZA is also a linear combination of 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 , so 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 is true.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtaUdGJDYlUScmIzk0NjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjdGN0Y3 Using algebra to rearrange terms we also computeLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=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\342\226\241But most subsets are not subspaces: (This was Exercise 5 yesterday):Exercise 1) Show that 1a) 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 is not a subspace of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjJGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 .1b) 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 is not a subspace of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjJGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 .1c) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZORlMtSShtZmVuY2VkR0YkNiYtRiM2MS1JJW1zdXBHRiQ2JS1GVjYmLUYjNiYtRiw2JVEieEYnRi9GMi1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIRlIvRk5RLDAuMzMzMzMzM2VtRictRiw2JVEieUYnRi9GMkY5RjkvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictRiM2JC1GLDYlUSJURidGL0YyRjkvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUY2Ni1RKiZFbGVtZW50O0YnRjlGO0Y+RkBGQkZERkZGSEZKRk0tRmVuNiUtRiw2JVEoJnJlYWxzO0YnL0YwRj1GOS1GIzYkLUkjbW5HRiQ2JFEiMkYnRjlGOUZicEZPLUYsNiVRInNGJ0YvRjItRjY2LVEiLkYnRjlGO0Y+RkBGQkZERkZGSEZSRlQtRiw2JVEidEYnRi9GMkZncUZPRmRvRjUtRmFxNiRRIjNGJ0Y5Rk9GW29GOUY5L0Zob1EifGZyRicvRltwUSJ8aHJGJ0Y5 is a subspace of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjJGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 . Then find a basis for this subspace. What is the dimension of this subspace?The geometry of subspaces in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSJtRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 is pretty special:Exercise 2) Use matrix theory to show that the only subspaces of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjJGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 are (0) The single vector LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtbkdGJDYkUSIwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjcvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGQC8lKnN5bW1ldHJpY0dGQC8lKGxhcmdlb3BHRkAvJS5tb3ZhYmxlbGltaXRzR0ZALyUnYWNjZW50R0ZALyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRidGM0Y3RjcvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictRiM2JS1JI21pR0YkNiVRIlRGJy8lJ2l0YWxpY0dGQy9GOFEnaXRhbGljRidGam5GXG8vJTFzdXBlcnNjcmlwdHNoaWZ0R0Y2Rjc= , or(1) A line through the origin, i.e. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVElc3BhbkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYkLUYsNihRInVGJy8lJWJvbGRHRjFGLy8lKnVuZGVybGluZUdGMS9GM1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRicvRjNRJ25vcm1hbEYnRkYvJSVvcGVuR1EifGZyRicvJSZjbG9zZUdRInxockYnRkY= for some non-zero vector LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2KFEidUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRicvRjdRJ25vcm1hbEYn , or(2) All of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjJGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 .Exercise 3) Can you use matrix theory to show that the only subspaces of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjNGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 are (0) The single vector 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 , or(1) A line through the origin, i.e. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVElc3BhbkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYkLUYsNihRInVGJy8lJWJvbGRHRjFGLy8lKnVuZGVybGluZUdGMS9GM1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRicvRjNRJ25vcm1hbEYnRkYvJSVvcGVuR1EifGZyRicvJSZjbG9zZUdRInxockYnRkY= for some non-zero vector LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2KFEidUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRicvRjdRJ25vcm1hbEYn , or(2) A plane through the origin, i.e.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 where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2KFEidUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSNtb0dGJDYtUSIsRicvRjdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNihRInZGJ0YvRjJGNEY2RjlGQA== are linearly independent, or(3) All of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjNGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 .Exercise 4) What are the dimensions of the subspaces in Exercise 2 and Exercise 3?Exercise 5) Yesterday we showed that the plane spanned by 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, 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 has implicit equation 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 .Suppose you were just given the implicit equation. How could you find a basis for this plane?This discussion leads toTheorem 2: Let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYnLUYvNiVRIm1GJ0YyRjUtSSNtb0dGJDYtUSgmdGltZXM7RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlQtRi82JVEibkYnRjJGNUYyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZB be a matrix. Consider the solution space LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= of all solutions to ths homogeneous matrix 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 ,i.e.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJXRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkobWZlbmNlZEdGJDYmLUYjNjEtRiw2KFEieEYnLyUlYm9sZEdGNkY0LyUqdW5kZXJsaW5lR0Y2L0Y4USxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJy1GOzYtUSomRWxlbWVudDtGJ0Y+RkBGQ0ZFRkdGSUZLRk1GT0ZSLUklbXN1cEdGJDYlLUYsNiVRKCZyZWFscztGJy9GNUZCRj4tRiM2JC1GLDYlUSJuRidGNEY3Rj4vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUY7Ni1RIn5GJ0Y+RkBGQ0ZFRkdGSUZLRk0vRlBRJjAuMGVtRicvRlNGZXAtRiw2JVEic0YnRjRGNy1GOzYtUSIuRidGPkZARkNGRUZHRklGS0ZNRmRwRmZwLUYsNiVRInRGJ0Y0RjdGanBGYXAtSSVtc3ViR0YkNiUtRiw2JVEiQUYnRjRGNy1GIzYmLUYsNiVRIm1GJ0Y0RjctRjs2LVEoJnRpbWVzO0YnRj5GQEZDRkVGR0ZJRktGTS9GUFEsMC4yMjIyMjIyZW1GJy9GU0ZfckZbcEY+LyUvc3Vic2NyaXB0c2hpZnRHRmBwRmFwRllGOi1JI21uR0YkNidGYHBGZm5GaG4vRjhGXm9GXG9GPkY+LyUlb3BlbkdRInxmckYnLyUmY2xvc2VHUSJ8aHJGJ0Y+RitGPg== .Then 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 is a subspace. Furthermore, you can find a basis for this subspace by backsolving from the reduced row echelon form of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=: once you write the solution in linear combination 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the vectors 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 will always be linearly independent. Thus, since these vectors span LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= by construction, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYuLUklbXN1YkdGJDYlLUkjbWlHRiQ2KFEidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lKnVuZGVybGluZUdGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2JC1JI21uR0YkNiRRIjFGJy9GOlEnbm9ybWFsRidGRS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiLEYnRkUvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGUC8lKnN5bW1ldHJpY0dGUC8lKGxhcmdlb3BHRlAvJS5tb3ZhYmxlbGltaXRzR0ZQLyUnYWNjZW50R0ZQLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRks2LVEifkYnRkVGTi9GUkZQRlNGVUZXRllGZW5GZ24vRltvRmluLUYsNiVGLi1GIzYkLUZCNiRRIjJGJ0ZFRkVGR0ZKRl1vRl1vLUZLNi1RIy4uRidGRUZORmBvRlNGVUZXRllGZW4vRmhuUSwwLjIyMjIyMjJlbUYnRmFvLUZLNi1RIi5GJ0ZFRk5GYG9GU0ZVRldGWUZlbkZnbkZhb0Zdby1GLDYlRi4tRiM2JC1GLzYlUSJrRidGNS9GOlEnaXRhbGljRidGRUZHRkU= are a basis for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.Exercise 6) Check that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtaUdGJDYlUScmIzk0NTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjdGNy1JI21vR0YkNi1RIixGJ0Y3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JC1GIzYkLUYxNiVRJyYjOTQ2O0YnRjRGN0Y3RjdGNw== hold for the solution space LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GLDYoUSJ4RicvJSVib2xkR0YxRi8vJSp1bmRlcmxpbmVHRjEvRjNRLGJvbGQtaXRhbGljRicvJStmb250d2VpZ2h0R1ElYm9sZEYnLUY2Ni1RIj1GJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjc3Nzc3OGVtRicvRk5GaW4tSSNtbkdGJDYnUSIwRidGUkZUL0YzRlpGWEY1LyUrZXhlY3V0YWJsZUdGMS8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RidGOQ== , i.e to verify 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 indeed a subspace.Illustrate how to find a basis for the solution space to a homogeneous matrix equation by completing this large example:Exercise 7) Consider the matrix equation 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 , with the matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (and its reduced row echelon form) shown below:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSShtYWN0aW9uR0YkNiQtRiM2Ji1GIzYmLUYjNiYtRiM2Ji1GIzYmLUYjNiYtRiM2Ji1GIzYmLUYjNiYtRiM2JS1JKG1mZW5jZWRHRiQ2KC1GIzYnLUknbXRhYmxlR0YkNjgtSSRtdHJHRiQ2Ky1JJG10ZEdGJDYoLUkjbW5HRiQ2JFEiMUYnRjkvJSlyb3dhbGlnbkdRIUYnLyUsY29sdW1uYWxpZ25HRmRwLyUrZ3JvdXBhbGlnbkdGZHAvJShyb3dzcGFuR0ZhcC8lK2NvbHVtbnNwYW5HRmFwLUZccDYoLUZfcDYkUSIyRidGOUZicEZlcEZncEZpcEZbcS1GXHA2KC1GX3A2JFEiMEYnRjlGYnBGZXBGZ3BGaXBGW3FGW3BGW3BGXXFGYnBGZXBGZ3AtRmlvNitGXXEtRlxwNigtRl9wNiRRIjRGJ0Y5RmJwRmVwRmdwRmlwRltxRltwRmlxRltwLUZccDYoLUZfcDYkUSI3RidGOUZicEZlcEZncEZpcEZbcUZicEZlcEZncC1GaW82Ky1GXHA2KC1GIzYoLUY2Ni1RKiZ1bWludXMwO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZdc0ZecC8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GJy8lKXJlYWRvbmx5R0YxLyUwZm9udF9zdHlsZV9uYW1lR1EqMkR+T3V0cHV0RidGOUZicEZlcEZncEZpcEZbcS1GXHA2KC1GIzYoRmlyRl9xRl9zRmJzRmRzRjlGYnBGZXBGZ3BGaXBGW3FGW3BGW3BGZ3NGW3BGYnBGZXBGZ3AtRmlvNitGZ3MtRlxwNigtRiM2KEZpckZbckZfc0Zic0Zkc0Y5RmJwRmVwRmdwRmlwRltxRmJxRmdzRmdzRl10RmJwRmVwRmdwLyUmYWxpZ25HUSVheGlzRicvRmNwUSliYXNlbGluZUYnL0ZmcFEmcmlnaHRGJy9GaHBRJ3xmcmxlZnR8aHJGJy8lL2FsaWdubWVudHNjb3BlR0YxLyUsY29sdW1ud2lkdGhHUSVhdXRvRicvJSZ3aWR0aEdGXnUvJStyb3dzcGFjaW5nR1EmMS4wZXhGJy8lLmNvbHVtbnNwYWNpbmdHUSYwLjhlbUYnLyUpcm93bGluZXNHUSVub25lRicvJSxjb2x1bW5saW5lc0dGaXUvJSZmcmFtZUdGaXUvJS1mcmFtZXNwYWNpbmdHUSwwLjRlbX4wLjVleEYnLyUqZXF1YWxyb3dzR0Y9LyUtZXF1YWxjb2x1bW5zR0Y9LyUtZGlzcGxheXN0eWxlR0Y9LyUlc2lkZUdGZ3QvJTBtaW5sYWJlbHNwYWNpbmdHRmZ1Rl9zRmJzRmRzRjlGOS9JK21zZW1hbnRpY3NHRiRRJ01hdHJpeEYnLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRlt3RjlGW3ctRiw2I0ZkcEY5Rlt3RmR3RjlGW3dGZHdGOUZbd0Zkd0Y5Rlt3RmR3RjlGW3dGZHdGOUZbd0Zkd0Y5Rlt3RmR3RjlGW3dGZHdGOUZbdy8lK2FjdGlvbnR5cGVHUS5ydGFibGVhZGRyZXNzRidGZHdGOQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEoJnNyYXJyO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDRi8=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7a) Find a basis for the solution space 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 by backsolving, writing your explicit solutions in linear combination form, and extracting a basis. Explain why these vectors span the solution space and verify that they're linearly indepdendent.7b) What is the dimension of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ? How is the dimension related to the shape of the reduced row echelon form of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=?7c) Since solutions to homogeneous matrix equations are exactly linear dependencies between the columns of the matrix, your basis in 7a can be thought of as a "basis" of the key column dependencies. Check this. (By the way, this connection between homogeneous solutions and column dependencies is the reason why any matrix has only one reduced row echelon form - the four conditions for rref (zero rows at bottom, non-zero rows start with leading ones, leading ones in successive rows move to the right, all column entries in any column containing a leading "1" are zero except for that "1"), together with specified column dependencies uniquely determine the rref matrix.)