MAT2705-01/02 14F Quiz 6 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 equal signs and arrows 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 should use technology for row reductions and determinants. Report results you find in this way: value of det, reduced matrix.]
1. a) Find the general solution of the following linear system, identifying your coefficient matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, the rhs vector 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 the augmented matrix, the rref matrix and give the solution in scalar form: 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.
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b) Write the solution in the vector form 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 and then express it as an explicit linear combination of a set of vectors (all but one of which have the free parameters as coefficients).c) Identify a basis of the solution space of the related homogeneous system corresponding to setting LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkmbW92ZXJHRiQ2JS1GIzYnLUkjbWlHRiQ2J1EiYkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNi8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRidGNEY3RjlGPC1JI21vR0YkNi5RLSZSaWdodEFycm93O0YnLyUscGxhY2Vob2xkZXJHRjYvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZ1bnNldEYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlgvRlVGNi1GQDYtUSI9RidGRS9GSFEmZmFsc2VGJy9GS0Zqbi9GTUZqbi9GT0Zqbi9GUUZqbi9GU0Zqbi9GVUZqbi9GV1EsMC4yNzc3Nzc4ZW1GJy9GWkZiby1GLDYlLUYjNictSSNtbkdGJDYmUSIwRidGNC9GOkY+RjxGNEY3RjlGPC1GQDYuRkJGQ0ZFRmluRltvRkxGXW9GXm9GX29GYG9GYW9GY29GZW5GRQ== . [Write your basis using curly brace set notation, using either angle bracket vector notation or column matrix notation.]d) Consider the set of vectors {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} which form the columns of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. What are the independent relationships among this vectors? (Write each such relationship in the form: a linear combination of them equals the zero vector.)e) What subset of these vectors does our solution algorithm show to be linearly independent automatically? Express LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21pR0YkNidRImJGJy8lJWJvbGRHUSV0cnVlRicvJSdpdGFsaWNHRjEvJSxtYXRodmFyaWFudEdRLGJvbGQtaXRhbGljRicvJStmb250d2VpZ2h0R1ElYm9sZEYnLUkjbW9HRiQ2LlEtJlJpZ2h0QXJyb3c7RicvJSxwbGFjZWhvbGRlckdGMS9GNVEnbm9ybWFsRicvJSZmZW5jZUdRJnVuc2V0RicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy9GUEYxas a unique linear combination of these latter vectors. [The particular solution!]
2. Which of the following sets of vectors are linearly independent? Justify your claim. Report the results of any calculation you use in doing so.
a) {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},
b) {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,
c) {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 .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1.Hmm, this one is right out of the book, and was assigned for HW.Qyc+SSJDRzYiLUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMvSSQlaWRHRiUiNTV2KVFQdVN1WSU9IiIiLUk2UmVkdWNlZFJvd0VjaGVsb25Gb3JtR0YlNiNGJEYvLy0mSSdWZWN0b3JHRig2I0knY29sdW1uR0YlNiMvRi0iNUl3KVFQdVN1WSU9LUkzQmFja3dhcmRTdWJzdGl0dXRlR0YlNiNJIiVHRiU=This is the vector form of the solution, but we would reverse the order of the parameters, working from top down instead of bottom up like Maple.
The general solution in scalar form (reversing order of parameters as we do by hand):
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Basis of the homogeneous solution space: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QyU+SSJBRzYiLUktRGVsZXRlQ29sdW1uR0YlNiRJIkNHRiU7IiIkIiImIiIiLUk2UmVkdWNlZFJvd0VjaGVsb25Gb3JtR0YlNiNJIiVHRiU=The leading columns (first two) are linearly independent and span the space of all the columns, giving a unique representation of the rhs vector:LywmLUknQ29sdW1uRzYiNiRJIkFHRiYiIiIiIiQtRiU2JEYoIiIjRiktRiU2JEkiQ0dGJiIiJg==There are two relationships among the columns:QyYvLCgtSSdDb2x1bW5HNiI2JEkiQUdGJyIiIiEiIy1GJjYkRikiIiNGLi1GJjYkSSJDR0YnIiIkRiotJkknVmVjdG9yRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YnNiNJJ2NvbHVtbkdGJzYjL0kkJWlkR0YnIjV5LSlRUHVTdVklPUYqLywoRiUhIiJGLEYyLUYmNiRGMSIiJUYqRjNGKg==check:LUkmc29sdmVHNiI2IzwlLywoJkkieEdGJDYjIiIiRiwmRio2IyIiJCIiIyZGKjYjIiIlRixGLy8sKkYpRiwmRio2I0YwRi9GLSEiJUYxISIpIiInLywqRilGMEY2IiIoRi0hIzVGMSEjPiIjOA==2.If the determinant is nonzero/zero they are linearly independent/dependent.
a)LUktUmFuZG9tTWF0cml4RzYiNiUiIiRGJi9JKmdlbmVyYXRvckdGJC1JJXJhbmRHRiQ2IzshIiYiIiY=QyctSSdNYXRyaXhHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy9JJCVpZEdGKCI1PTYnUVB1U3VZJT0iIiItSSxEZXRlcm1pbmFudEdGKDYjSSIlR0YoRi0tSTZSZWR1Y2VkUm93RWNoZWxvbkZvcm1HRig2I0kjJSVHRig=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 independent.
b)QyUtSS1SYW5kb21NYXRyaXhHNiI2JSIiJEYnL0kqZ2VuZXJhdG9yR0YlLUklcmFuZEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjOyEiJEYnIiIiLUksRGV0ZXJtaW5hbnRHRiU2I0kiJUdGJQ==QyUtSSdNYXRyaXhHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy9JJCVpZEdGKCI1USkpKlFQdVN1WSU9IiIiLUksRGV0ZXJtaW5hbnRHRig2I0kiJUdGKA==QyUtSSQ8fGdyPkdJKF9zeXNsaWJHNiI2JS1JJDwsPkdGJTYlIiIhISIjISIiLUYpNiVGLSIiIkYtLUYpNiUiIiMhIiVGMEYwLUksRGV0ZXJtaW5hbnRHRiY2I0kiJUdGJg==JSFHLinearly dependent.c)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