Homework 6: Linear Independence (Part 2)

Instructions. Assignments should be stapled and written clearly and legibly.

1. Section 4.3, #5(a), 12, 15(a)
2. Determine whether \(\{1, \ln(2x),\ln(x^2)\}\) is linearly independent in \(F(0,\infty)\). Justify your answer (a) by using the definition of linear independence/linear dependence from class, and (b) by using Theorem 1.4 from the study guide.
3. Recall that in class we proved \(\{ \cos^2 x,\sin^2 x, \cos 2x \} \) is linearly dependent in \( F(-\infty,\infty) \) by using a trigonometric identity. Prove that, however, \(\{ \cos^2 x,\sin^2 x, \cos x \} \) is linearly independent in \( F(-\infty,\infty) \). Use the following strategy:
   1. Assume that $$a \cos^2 x + b \sin^2 x + c \cos x = 0$$ in \(F(-\infty,\infty) \) for some scalars \(a, b, c\).
   2. Substituting any value for \(x\) into the above equation will produce a linear equation in \(a\), \(b\), and \(c\). Choose three values of \(x\) so that the three produced linear equations result in a linear system with a unique solution. (I recommend that you choose values of \(x\) for which \(\cos x\) and \(\sin x\) are both as simple as possible.)
   3. Explain why your work in (b) implies that \(\{ \cos^2 x,\sin^2 x, \cos x \} \) is linearly independent in \( F(-\infty, \infty) \).
4. Let \(\bv_1,\ldots,\bv_p\) be vectors in a vector space \(V\). Prove the following:
   1. Let \(\bv_1,\ldots,\bv_p\) span the vector space \(V\), and let \(\bu\) be any vector in \(V\). Then \(\{\bu,\bv_1,\ldots,\bv_p\}\) is linearly dependent.
   2. Let \(\{\bv_1,\bv_2,\ldots,\bv_p\}\) be linearly independent. Then \(\bv_2,\ldots,\bv_p\) cannot span \(V\).
   Hint. Use Theorem 1.4 from the study guide for both parts.