1.7soln

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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online