Homework 5: Linear Independence

Instructions. In Problems 2 and 3, make sure to use the study guide's definition of linear independence. It is different from the textbook's definition.

1. Section 4.3, #3(a), 4(a), 15(b), 16(a). Try to solve 15(b) and 16(a) with as few computations as possible.
2. Let \(\{\bv_1,\bv_2,\bv_3,\bv_4\}\) be a linearly independent set of vectors in a vector space \(V\). Using only the definition of linear independence, prove that \(\{\bv_1,\bv_2,\bv_3\}\) is linearly independent as well.
   Hint. Begin the proof as follows: ``Suppose that \(k_1\bv_1+k_2\bv_2+k_3\bv_3=\mathbf{0}\) for some scalars \(k_1, k_2, k_3\). I must show that \(k_1=k_2=k_3=0\).''
3. Using only the definition of linear independence, prove that if \(\{\bu,\bv,\bw\}\) is linearly independent, then so is \(\{\bu+\bv,\bu+\bw,\bv+\bw\}\).
   Hint. Begin the proof as follows: ``Suppose that \(c(\bu+\bv)+ d(\bu+\bw)+ e(\bv+\bw)=\mathbf{0}\) for some scalars \(c, d, e\). I must show that \(c=d=e=0\).''