MAT2705-03/04 11F Test 2 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, determinants and matrix inverses.
1. a) On the grid to the right, draw in arrows representing the vectors 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 and 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 and label them by their symbols. Then draw in the parallelogram that graphically expresses LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbW92ZXJHRiQ2JS1GIzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiM0YnL0Y7USdub3JtYWxGJ0ZDLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSFGJ0ZDLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZOLyUpc3RyZXRjaHlHRk4vJSpzeW1tZXRyaWNHRk4vJShsYXJnZW9wR0ZOLyUubW92YWJsZWxpbWl0c0dGTi8lJ2FjY2VudEdGTi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRmduRkMtRkk2LlEtJlJpZ2h0QXJyb3c7RicvJSxwbGFjZWhvbGRlckdGOUZDL0ZNUSZ1bnNldEYnL0ZQRmBvL0ZSRjkvRlRGYG8vRlZGYG8vRlhGYG8vRlpGYG8vRmZuUSYwLjBlbUYnL0ZpbkZoby9GWkY5RkhGQw== as a linear combination of 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. Label its two sides that intersect at the origin by the corresponding vectors they represent. Extend the basis vectors 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 to the corresponding coordinate axes for 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 and mark the positive direction with an arrow head and the axis label.
From the grid, read off the coordinates 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of 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 with respect to these two vectors (write them down) and express 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 as a linear combination of these vectors; put this equation at the tip of this vector. Explain how you got these numbers.b) Now write down the matrix equation that enables you to express 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 as a linear combination of the other two vectors, solve that system using matrix methods, and then express 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 explicitly as a linear combination of those vectors.
c) Check your linear combination by expanding it out to get the original vector. Did you?
d) Does your matrix result agree with part 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6"
2. For LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JKG1mZW5jZWRHRiQ2Ji1GIzYrLUklbXN1YkdGJDYlLUYjNictRjY2LVEhRidGOUY7Rj5GQEZCRkRGRkZIRkpGTS1GIzYlRlktSSZtb3ZlckdGJDYlLUYsNiVRInZGJ0YvRjItRjY2LlEtJlJpZ2h0QXJyb3c7RicvJSxwbGFjZWhvbGRlckdGMUY5L0Y8USZ1bnNldEYnL0Y/RmRvL0ZBRjEvRkNGZG8vRkVGZG8vRkdGZG8vRklGZG8vRktRJjAuMGVtRicvRk5GXHAvRklGMUY5RllGL0YyLUYjNiUtSSNtbkdGJDYkUSIxRidGOUYvRjIvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GNjYtUSJ8Z3JGJ0Y5RjtGPkZmb0ZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnL0ZORlxxLUZVNiVGVy1GIzYlLUZicDYkUSIyRidGOUYvRjJGZXBGaHAtRlU2JUZXLUYjNiUtRmJwNiRRIjNGJ0Y5Ri9GMkZlcEZocC1GVTYlRmhuLUYjNiYtRmJwNiRRIjRGJ0Y5LyUrZXhlY3V0YWJsZUdGMS8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RidGOUZlcEZZRjlGOS8lJW9wZW5HUScmbGFuZztGJy8lJmNsb3NlR1EnJnJhbmc7RidGOQ== with 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,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, 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.
a) Does 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 lie in the span of this set? If so, show how it can be expressed in terms of them in the most general way. If not, show why not. Show all work to support your claim.
b) Does 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 lie in the span of this set? If so, show how it can be expressed in terms of them in the most general way. If not, show why not. Show all work to support your claim.c) Solve the equations 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 (state these matrix equations first), writing down the augmented matrix and its RREF form, identifying Leading and Free variables, and stating your result for 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.d) From your general solution write down a basis 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 of the solution space (in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjRGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIUYnRjUvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZZRjU=), using proper notation for a basis of vectors (as a set) in horizontal vector format LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiktSSNtaUdGJDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GOzYtUSJ+RidGPkZAL0ZERkJGRUZHRklGS0ZNRk8vRlNGUS1GOzYtUSMuLkYnRj5GQEZYRkVGR0ZJRktGTS9GUFEsMC4yMjIyMjIyZW1GJ0ZZLUY7Ni1RIi5GJ0Y+RkBGWEZFRkdGSUZLRk1GT0ZZRlVGPkY+LyUlb3BlbkdRJyZsYW5nO0YnLyUmY2xvc2VHUScmcmFuZztGJ0Y+.e) Write down the independent linear relationships among the original vectors that correspond to this basis. f) Does the span of the original set of vectors 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 represent a line, plane, a hyperplane or all of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjRGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1? Explain. State a basis for this subspace of LUklbXN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJnJlYWxzO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSVtcm93R0YkNiQtSSNtbkdGJDYkUSI0RidGMkYyLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJw== (which might be all of LUklbXN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJnJlYWxzO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSVtcm93R0YkNiQtSSNtbkdGJDYkUSI0RidGMkYyLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJw==), using proper notation for a basis of vectors (as a set) in horizontal vector format 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.
Sign and date pledge on reverse side at the end of your exam.solution1. combining vectors in the planeb)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==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c)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)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sides are respectively -1 and 2 times the basis vectors, so the new coordinates are (-1,2).2. span, linear 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No solution, so this vector cannot be expressed as a linear combination of the other 4.b) first we create a vector which does lie in the span: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and now we use it.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c) Since there are two free variables, there are nonzero solutions to the homogeneous system, so the 4 vectors are not linearly independent.d)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEkQXVnRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEqJmNvbG9uZXE7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSShtZmVuY2VkR0YkNiYtRiM2KC1GLDYlUSJBRidGL0YyLUY2Ni1RInxnckYnRjlGO0Y+L0ZBRjFGQkZERkZGSC9GS1EsMC4xMTExMTExZW1GJy9GTkZmbi1GLDYlUStaZXJvTWF0cml4RidGL0YyLUZQNiQtRiM2Jy1JI21uR0YkNiRRIjRGJ0Y5LUY2Ni1RIixGJ0Y5RjsvRj9GMUZARkJGREZGRkgvRktRJjAuMGVtRicvRk5RLDAuMzMzMzMzM2VtRictRmBvNiRRIjFGJ0Y5LyUrZXhlY3V0YWJsZUdGPUY5RjlGXnBGOUY5LyUlb3BlbkdRJyZsYW5nO0YnLyUmY2xvc2VHUScmcmFuZztGJ0ZecEY5LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkjbWlHRiQ2JVE2UmVkdWNlZFJvd0VjaGVsb25Gb3JtRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEkQXVnRidGL0YyL0YzUSdub3JtYWxGJ0Y9LUkjbW9HRiQ2LVEiO0YnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdGVC8lJmRlcHRoR0Znbi8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GLDYlUSRzb2xGJ0YvRjItRkA2LVEqJmNvbG9uZXE7RidGPUZDL0ZHRkVGSEZKRkxGTkZQL0ZTRldGVS1GQDYtUSJ+RidGPUZDRmVvRkhGSkZMRk5GUEZSL0ZWRlQtRiw2JVEzQmFja3dhcmRTdWJzdGl0dXRlRidGL0YyLUY2NiQtRiM2JC1GLDYlUSIlRidGL0YyRj1GPS8lK2V4ZWN1dGFibGVHRkVGPQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUklbXN1YkdGJDYlLUYjNictSSNtb0dGJDYtUSFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjkvJSlzdHJldGNoeUdGOS8lKnN5bW1ldHJpY0dGOS8lKGxhcmdlb3BHRjkvJS5tb3ZhYmxlbGltaXRzR0Y5LyUnYWNjZW50R0Y5LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSC1GIzYlRjAtSSZtb3ZlckdGJDYlLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnL0Y1USdpdGFsaWNGJy1GMTYuUS0mUmlnaHRBcnJvdztGJy8lLHBsYWNlaG9sZGVyR0ZWRjQvRjhRJnVuc2V0RicvRjtGaW4vRj1GVi9GP0Zpbi9GQUZpbi9GQ0Zpbi9GRUZpbi9GR1EmMC4wZW1GJy9GSkZhby9GRUZWRjRGMEZURlctRiM2JS1JI21uR0YkNiRRIjFGJ0Y0RlRGVy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUYxNi1RKiZjb2xvbmVxO0YnRjRGN0Y6RjxGPkZARkJGREZGRkktSShtZmVuY2VkR0YkNiYtRiM2LS1GMTYtUSomdW1pbnVzMDtGJ0Y0RjdGOkY8Rj5GQEZCRkQvRkdRLDAuMjIyMjIyMmVtRicvRkpGaXBGZm8tRjE2LVEiLEYnRjRGNy9GO0ZWRjxGPkZARkJGREZgby9GSlEsMC4zMzMzMzMzZW1GJ0ZlcC1GZ282JFEiMkYnRjRGW3FGZm9GW3EtRmdvNiRGXHBGNC8lK2V4ZWN1dGFibGVHRjlGNEY0LyUlb3BlbkdRJyZsYW5nO0YnLyUmY2xvc2VHUScmcmFuZztGJy1GMTYtUSI7RidGNEY3Rl5xRjxGPkZARkJGREZgb0ZJLUYxNi1RIn5GJ0Y0RjdGOkY8Rj5GQEZCRkRGYG9GYm8tRiw2JS1GIzYmRjBGS0YwRjQtRiM2JEZhcUY0RmpvRl1wLUZhcDYmLUYjNitGZXAtRmdvNiRRIjNGJ0Y0RltxRmFxRltxRmRxRltxRmZvRjRGNEZocUZbckYwRmZxRjQ=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEkc29sRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZMLUkobWZlbmNlZEdGJDYkLUYjNistSSVtc3ViR0YkNiUtRiw2JVEkX3QyRidGL0YyLUYjNigtSSNtbkdGJDYkUSIyRidGOS8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GJy8lK2V4ZWN1dGFibGVHRj0vJSlyZWFkb25seUdGMS8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnRjkvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZORltwLUZVNiUtRiM2Ji1GNjYtUSFGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjc3Nzc3OGVtRicvRk5GZXAtRiM2JUZhcC1JJm1vdmVyR0YkNiUtRiw2JVEidUYnRi9GMi1GNjYuUS0mUmlnaHRBcnJvdztGJy8lLHBsYWNlaG9sZGVyR0YxRjkvRjxRJnVuc2V0RicvRj9GZXEvRkFGMS9GQ0ZlcS9GRUZlcS9GR0ZlcS9GSUZlcUZqb0ZccC9GSUYxRjlGYXBGOS1GIzYkLUZnbjYkUSIxRidGOUY5RmRvLUY2Ni1RIitGJ0Y5RjtGPkZARkJGREZGRkhGSkZNLUZVNiVGVy1GIzYoRl9yRmpuRl1vRl9vRmFvRjlGZG9GZ28tRlU2JUZfcC1GIzYkRmZuRjlGZG9GXW9GOUY5Rl1vRjk=The two independent relationships are respectively: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 there are 2 relationships among 4 vectors only 2 are independent, so this set of vectors spans a plane in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiNEYnL0Y2USdub3JtYWxGJ0YyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRj4=. The first two vectors of the set form a basis: NiI=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEifGZyRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSV0cnVlRicvJSpzZXBhcmF0b3JHUSZmYWxzZUYnLyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjcvJShsYXJnZW9wR0Y3LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lJ2xzcGFjZUdRLDAuMTY2NjY2N2VtRicvJSdyc3BhY2VHRkRGLw==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,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
(or indeed any two of the 4 vectors, since no two are proportional) form a basis of this plane.JSFHpledgeWhen you have completed the exam, please read and sign the dr bob integrity pledge and hand this test sheet in on top of your answer sheets as a cover page, with the first test page facing up:"During this examination, all work has been my own. I give my word that I have not resorted to any ethically questionable means of improving my grade or anyone else's on this examination and that I have not discussed this exam with anyone other than my instructor, nor will I until after the exam period is terminated for all participants."
Signature: Date: TTdSMApJNlJUQUJMRV9TQVZFLzE2NTIzNzkwNFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiUiJSIiIiIiJCIiIyEiIkYlTTdSMApJNlJUQUJMRV9TQVZFLzQ1NDA3MTU1MlgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiUiJSIiIyIiIkYoIiIhRiU=TTdSMApJNlJUQUJMRV9TQVZFLzQ1NDA3MTY4OFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiUiJSIiJkYnIiIlISIiRiU=TTdSMApJNlJUQUJMRV9TQVZFLzQ1NDA3MTgyNFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiUiJSEiIiIiKCIiJSEiJEYlTTdSMApJNlJUQUJMRV9TQVZFLzQ1NDEwNDczNlgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiUiJSIiKSIiKiIiKCIiIUYlTTdSMApJNlJUQUJMRV9TQVZFLzQ1NDEzNzY1NlgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIzEiJSIlIiIiIiIkIiIjISIiRilGJ0YnIiIhIiImRiwiIiVGKkYqIiIoRi0hIiRGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzQ1NDE3Mjg0MFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiUiJSYlInhHNiMiIiImRig2IyIiIyZGKDYjIiIkJkYoNiMiIiVGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzQ1NDE3Mjk3NlgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiUiJSwqJiUieEc2IyIiIkYrJkYpNiMiIiNGLiZGKTYjIiIkIiImJkYpNiMiIiUhIiIsKkYoRjFGLEYrRi9GMkYzIiIoLCpGKEYuRixGK0YvRjVGM0Y1LChGKEY2Ri9GNkYzISIkRiU=TTdSMApJNlJUQUJMRV9TQVZFLzQ1NDEwNDczNlgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiUiJSIiKSIiKiIiKCIiIUYlTTdSMApJNlJUQUJMRV9TQVZFLzQ1NDE3MzI0OFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIzUiJSImIiIiIiIkIiIjISIiRilGJ0YnIiIhIiImRiwiIiVGKkYqIiIoRi0hIiQiIikiIipGLkYrRiU=TTdSMApJNlJUQUJMRV9TQVZFLzQ1NDE3MzU3NlgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIzUiJSImIiIiIiIhRihGKEYoRidGKEYoRiciIiNGKEYoIiIkISIjRihGKEYoRihGJ0YoRiU=TTdSMApJNlJUQUJMRV9TQVZFLzQ1NDIwNjU1MlgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiUiJSIiKSIiKiIiKCEiI0YlTTdSMApJNlJUQUJMRV9TQVZFLzQ1NDIwNjY4OFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiUiJSwqJiUieEc2IyIiIkYrJkYpNiMiIiNGLiZGKTYjIiIkIiImJkYpNiMiIiUhIiIsKkYoRjFGLEYrRi9GMkYzIiIoLCpGKEYuRixGK0YvRjVGM0Y1LChGKEY2Ri9GNkYzISIkRiU=TTdSMApJNlJUQUJMRV9TQVZFLzQ1NDIwNjU1MlgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiUiJSIiKSIiKiIiKCEiI0YlTTdSMApJNlJUQUJMRV9TQVZFLzQ1NDIwNTc5MlgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIzUiJSImIiIiIiIkIiIjISIiRilGJ0YnIiIhIiImRiwiIiVGKkYqIiIoRi0hIiQiIikiIipGLiEiI0YlTTdSMApJNlJUQUJMRV9TQVZFLzQ1NDIwNjEyMFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIzUiJSImIiIiIiIhRihGKEYoRidGKEYoRiciIiNGKEYoIiIkISIjRihGKEYpRipGKEYoRiU=TTdSMApJNlJUQUJMRV9TQVZFLzQ1NDE3Mjg0MFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiUiJSYlInhHNiMiIiImRig2IyIiIyZGKDYjIiIkJkYoNiMiIiVGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzQ1NDIwNTY1NlgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiUiJSwoIiIjIiIiJiUkX3QwRzYjRighIiImRis2I0YpISIkLCgiIiRGKUYqISIjRi5GKEYqRi5GJQ==TTdSMApJNlJUQUJMRV9TQVZFLzQ1NDIwNjU1MlgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiUiJSIiKSIiKiIiKCEiI0YlTTdSMApJNlJUQUJMRV9TQVZFLzQ1NDIwNzk5MlgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiUiJSIiKSIiKiIiKCEiI0YlTTdSMApJNlJUQUJMRV9TQVZFLzQ1NDIwNzA5NlgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIzUiJSImIiIiIiIkIiIjISIiRilGJ0YnIiIhIiImRiwiIiVGKkYqIiIoRi0hIiRGK0YrRitGK0YlTTdSMApJNlJUQUJMRV9TQVZFLzQ1NDIwOTE2MFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIzUiJSImIiIiIiIhRihGKEYoRidGKEYoRiciIiNGKEYoIiIkISIjRihGKEYoRihGKEYoRiU=TTdSMApJNlJUQUJMRV9TQVZFLzQ1NDIwNzQyNFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiUiJSwmJiUkX3QyRzYjIiIjISIiJkYpNiMiIiIhIiQsJkYoISIjRi1GK0YoRi1GJQ==TTdSMApJNlJUQUJMRV9TQVZFLzQ1NDIwOTQ4OFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiUiJSEiIiEiIyIiIiIiIUYlTTdSMApJNlJUQUJMRV9TQVZFLzQ1NDIwOTYyNFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiUiJSEiJCIiIyIiISIiIkYlTTdSMApJNlJUQUJMRV9TQVZFLzQ1NDI0MjUyMFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiUiJSIiIUYnRidGJ0YlTTdSMApJNlJUQUJMRV9TQVZFLzQ1NDI3NDUzNlgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiUiJSIiIUYnRidGJ0YlTTdSMApJNlJUQUJMRV9TQVZFLzQ1NDI3NDY3MlgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiUiJSIiIUYnRidGJ0Yl