LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnMAT2705-01/04 07S Quiz 5 Print Name (Last, First) _________________________________|__Show all work, including mental steps, in a clearly organized way that speaks for itself. Use proper mathematical notation, identifying expressions by their proper symbols (introducing them if necessary), and use arrows and equal signs when appropriate. Always simplify expressions. BOX final short answers. LABEL parts of problem. Keep answers EXACT (but give decimal approximations for interpretation). Indicate where technology is used and what type (Maple, GC).[You may use technology for row reductions, backsubstitutions and determinants although these may easily be done by hand.]
1. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYzLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2MFEiPUYnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkkvJSlzdHJldGNoeUdGSS8lKnN5bW1ldHJpY0dGSS8lKGxhcmdlb3BHRkkvJS5tb3ZhYmxlbGltaXRzR0ZJLyUnYWNjZW50R0ZJLyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0Zlbi8lKG1pbnNpemVHRj0vJShtYXhzaXplR1EpaW5maW5pdHlGJy1GRDYwUSJ+RidGPkZHRkpGTEZORlBGUkZUL0ZXUSFGJy9GWlEkMGVtRicvRmduRmNvRmhuRmpuLUkobWZlbmNlZEdGJDYmLUYjNilGOi1GRDYwUSIsRidGPkZHL0ZLRjRGTEZORlBGUkZURlZGYm8vRmduUTN2ZXJ5dGhpY2ttYXRoc3BhY2VGJ0ZobkZqbkY6RmpvRjpGam9GOkY+LyUlb3BlbkdRJyZsYW5nO0YnLyUmY2xvc2VHUScmcmFuZztGJ0Zqby1GLDYlRi4tRiM2Iy1GOzYkUSIyRidGPkZARkMtRmZvNiYtRiM2KUY6RmpvLUY7NiRRIjBGJ0Y+RmpvRjpGam9GYXFGPkZgcEZjcEZqby1GLDYlRi4tRiM2Iy1GOzYkUSIzRidGPkZARkMtRmZvNiYtRiM2KUZqcEZqb0Y6RmpvRmpwRmpvRjpGPkZgcEZjcEZqby1GLDYlRi4tRiM2Iy1GOzYkUSI0RidGPkZARkMtRmZvNiYtRiM2KUZqcEZqb0ZocUZqb0ZqcEZqb0ZocUY+RmBwRmNwRmpvLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiNUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2MFEiPUYnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkkvJSlzdHJldGNoeUdGSS8lKnN5bW1ldHJpY0dGSS8lKGxhcmdlb3BHRkkvJS5tb3ZhYmxlbGltaXRzR0ZJLyUnYWNjZW50R0ZJLyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0Zlbi8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJy1GRDYwUSJ+RidGPkZHRkpGTEZORlBGUkZUL0ZXUSFGJy9GWlEkMGVtRicvRmduRmRvRmhuRltvLUkobWZlbmNlZEdGJDYmLUYjNiktRjs2JEZqbkY+LUZENjBRIixGJ0Y+RkcvRktGNEZMRk5GUEZSRlRGVkZjby9GZ25RM3Zlcnl0aGlja21hdGhzcGFjZUYnRmhuRltvLUY7NiRRIjJGJ0Y+Rl1wRltwRl1wRmNwRj4vJSVvcGVuR1EnJmxhbmc7RicvJSZjbG9zZUdRJyZyYW5nO0YnLUZENjBRIi5GJ0Y+RkdGSkZMRk5GUEZSRlRGVkZjb0Zlb0ZobkZbbw==a) Express LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiNUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn as a linear combination of the remaining 4 vectors, in the most general way. [Final answer: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiNUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2MFExJkludmlzaWJsZVRpbWVzO0YnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkkvJSlzdHJldGNoeUdGSS8lKnN5bW1ldHJpY0dGSS8lKGxhcmdlb3BHRkkvJS5tb3ZhYmxlbGltaXRzR0ZJLyUnYWNjZW50R0ZJLyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSQwZW1GJy8lJ3JzcGFjZUdGZW4vJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5RictRkQ2MFEiPUYnRj5GR0ZKRkxGTkZQRlJGVEZWL0ZaUS90aGlja21hdGhzcGFjZUYnL0ZnbkZib0ZobkZbby1GRDYwUSJ+RidGPkZHRkpGTEZORlBGUkZUL0ZXUSFGJ0ZZRmZuRmhuRltvLUZENjBRIy4uRidGPkZHRkpGTEZORlBGUkZUL0ZXUShwb3N0Zml4RicvRlpRMG1lZGl1bW1hdGhzcGFjZUYnRmZuRmhuRltvLUZENjBRIi5GJ0Y+RkdGSkZMRk5GUEZSRlRGVkZZRmZuRmhuRltvLUZENjBRIl1GJ0Y+L0ZIRjRGSi9GTUY0Rk5GUEZSRlRGXHAvRlpRLnRoaW5tYXRoc3BhY2VGJy9GZ25RMnZlcnl0aGlubWF0aHNwYWNlRidGaG5GW28=
b) Check that this general linear combination that you find actually evaluates to 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
c) Find the independent linear relationships among these 4 vectors. Write out these relationships individually.
OPTIONAL. If you have time while others are finishing up, try this:
2. Demonstrate that the following vectors are linearly independent:
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.solutionQyZJKHJlc3RhcnRHJSpwcm90ZWN0ZWRHISIiLUkld2l0aEc2IjYjJkkoU3R1ZGVudEdJKF9zeXNsaWJHRig2I0kuTGluZWFyQWxnZWJyYUc2JEYsRiRGJQ==1.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a) 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a)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtRiw2JVE2UmVkdWNlZFJvd0VjaGVsb25Gb3JtRidGL0YyLUkjbW9HRiQ2MFEwJkFwcGx5RnVuY3Rpb247RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJWZvcm1HUSZpbmZpeEYnLyUnbHNwYWNlR1EkMGVtRicvJSdyc3BhY2VHRlQvJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5RictSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSJDRidGL0YyRj4tRjs2MFEiO0YnRj5GQC9GREYxRkVGR0ZJRktGTUZPRlIvRlZRL3RoaWNrbWF0aHNwYWNlRidGV0ZaLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjBlbUYnLyUmZGVwdGhHRmpvLyUqbGluZWJyZWFrR1EobmV3bGluZUYnRistRiM2KC1GLDYlUSJjRidGL0YyLUY7NjBRIzo9RidGPkZARkNGRUZHRklGS0ZNRk8vRlNGZG9GY29GV0ZaLUYsNiVRM0JhY2t3YXJkU3Vic3RpdHV0ZUYnRi9GMkY6LUZobjYkLUYjNiMtRjs2MFEiJUYnRj5GQEZDRkVGR0ZJRktGTUZPRltxRmNvRldGWkY+RitGKw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNidGKy1GIzYjLUYsNiVRMUxpbmVhclNvbHZlVHV0b3JGJ0YvRjItSSNtb0dGJDYwUTAmQXBwbHlGdW5jdGlvbjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSQwZW1GJy8lJ3JzcGFjZUdGVi8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRIkNGJ0YvRjJGQEYrRis=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Ynb)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c)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2.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnOne may evaluate the determinant to show that it is nonzero, which is equivalent to showing that the matrix row reduces to the identity matrix.LUksRGV0ZXJtaW5hbnRHNiQvSSttb2R1bGVuYW1lRzYiSS5MaW5lYXJBbGdlYnJhRzYkSShfc3lzbGliR0YnJSpwcm90ZWN0ZWRHRis2I0kiVUdGJw==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Two addrow operations, a swap and a final addrow operation reduces this to upper triangular form.Minus the product of the diagonal values is the determinant:LCQqJiMiIzUiIiQiIiItSSIqRyUqcHJvdGVjdGVkRzYkLUYpNiRGJyEiJiEiJ0YnISIiLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn