Homework 7: Basis and Coordinates

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

1. Section 4.4, #19(a)(b), 13(b), 14(b), 24(a)(b), 27(a)(b).
2. Suppose that \(\bu\), \(\bv\), and \(\bw\) are vectors in a vector space \(V\), and suppose that none of the three vectors \(\bu\), \(\bv\), or \(\bw\) is a scalar multiple of either of the other two vectors. In other words, \(\bu\) is not a scalar multiple of \(\bv\) or of \(\bw\); \(\bv\) is not a scalar multiple of \(\bu\) or of \(\bw\); and \(\bw\) is not a scalar multiple of \(\bu\) or of \(\bv\). Is \(\{\bu, \bv, \bw\}\) necessarily linearly independent? If so, prove it. If not, give a concrete counterexample.
3. Suppose that \(\cB=\{\bv_1,\ldots,\bv_n\}\) is a linearly idependent set of vectors in a vector space \(V\). Explain how we know that \(\sspan \cB\) is a subspace of \(V\). Then prove that \(\cB\) is a basis for \(\sspan \cB\).
4. In this problem you will prove that \(\{1,2^x, 3^x\}\) is a basis for \(\sspan\{1,2^x, 3^x\}\).
   1. Prove that \(\{1,2^x, 3^x\}\) is linearly independent in \(F(-\infty, \infty)\). Use the method outlined in Problem 3 of Homework 6.
   2. Explain how we know that \(\sspan\{1,2^x, 3^x\}\) is a subspace of \(F(-\infty, \infty)\).
   3. Use Problem 3 above to prove that \(\{1,2^x, 3^x\}\) is a basis for \(\sspan\{1,2^x, 3^x\}\).
5. In Problem 2 of Homework 6, you found that \(\{1, \ln(2x), \ln(x^2)\}\) is linearly dependent in \(F(0,\infty)\). Let \(W=\sspan\{1, \ln(2x), \ln(x^2)\}\).
   1. Explain how we know that \(W\) is a subspace of \(F(0,\infty)\)?
   2. Find two separate bases \(\cB\) and \(\cC\) for \(W\), and find \(\dim W\). Make sure to verify that each of your two asserted bases is linearly independent, and then use Problem 3 to justify that each is a basis.
   3. Determine whether \(\ln (7x^{12})\) is in \(W\). If so, find \([\ln(7x^{12})]_{\cB}\) and \([\ln(7x^{12})]_{\cC}\), where \(\cB\) and \(\cC\) are the bases you found in part (b).
   You should justify your answers, but formal proofs are not required for this problem.