1.7soln - Solutions to problems from section 1.7 0 3 0 2....

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Solutions to problems from section 1.7 2. Determine if the vectors 0 0 2 , 0 5 - 8 , - 3 4 1 are linearly independent. Solution: Consider the matrix A whose columns are the vectors listed above. The vectors above are linearly independent if and only if the equation A x = 0 has only the trivial solution, which happens if and only if the matrix A has a pivot position in each column (otherwise there would be free variables). Well, A = 0 0 - 3 0 5 4 2 - 8 1 R 1 R 3 -→ 2 - 8 1 0 5 4 0 0 -3 has a pivot position in each column, thus the vectors 0 0 2 , 0 5 - 8 , - 3 4 1 are linearly independent. 6. Determine if the columns of the matrix 4 - 3 0 0 - 12 4 1 0 3 5 4 6 are linearly independent. Solution: Using the same argument as in exercise 2 we just need to check whether or not the matrix above has a pivot position in each column: 4 - 3 0 0 - 12 4 1 0 3 5 4 6 R 1 R 3 -→ 1 0 3 0 - 12 4 4 - 3 0 5 4 6 NR 3 = R 3 - 4 R 1 NR 4 = R 4 - 5 R 1 -→ 1 0 3 0 - 12 4 0 - 3 - 12 0 4 - 6 R 2 R 3 -→ 1 0 3 0 - 3 - 12 0 - 12 4 0 4 - 6 NR 2 = - 1 3 R 3 -→ 1 0 3 0 1 4 0 - 12 4 0 4 - 6 NR 3 = R 3 +12 R 2 NR 4 = R 4 - 4 R 2 -→ 1 0 3 0 1 4 0 0 16 0 0 - 22 NR 4 = R 4 + 22 16 R 3 -→ 1
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1.7soln - Solutions to problems from section 1.7 0 3 0 2....

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