Tiêu đề Linear dependence and independence_01.pdf ✅

Linear Dependence and Independence Page | 23 Linear Dependence and Linear Independence Definition 10.2.4. A set S ൌ ሼvଵ, vଶ,..., v௡ሽ of vectors in a vector space V is called linearly independent if the following vector equation cଵvଵ ൅ cଶvଶ ൅⋯൅c௡v௡ ൌ 0 has the only trivial solution cଵ ൌ cଶ ൌ⋯ൌc௡ ൌ 0. If the vector equation has any nontrivial solution, then S is called linearly dependent. In other words, S is called linearly independent if none of the vectors in S is a linear combination of other vectors in S. Otherwise, S is called linearly dependent. Example 10.2.6. Show that Sൌ൝൥ 1 0 0 ൩, ൥ 0 1 0 ൩, ൥ 0 0 1 ൩ൡ is linearly independent. Solution. Consider the following equation cଵvଵ ൅ cଶvଶ ൅ cଷvଷ ൌ 0 cଵ ൥ 1 0 0 ൩ ൅ cଶ ൥ 0 1 0 ൩ ൅ cଷ ൥ 0 0 1 ൩ ൌ ൥ 0 0 0 ൩ ൥ cଵ cଶ cଷ ൩ ൌ ൥ 0 0 0 ൩ Therefore, cଵ ൌ cଶ ൌ cଷ ൌ 0. Hence S is linearly independent. Example 10.2.7. Show that S ൌ ⎩ ⎪ ⎨ ⎪ ⎧ ൥ 1 2 3 ൩ ถvభ , ൥ 0 1 2 ൩ ถvమ , ൥ െ2 0 1 ൩ ถvయ ⎭ ⎪ ⎬ ⎪ ⎫ is linearly independent. Solution. Consider the following equation cଵvଵ ൅ cଶvଶ ൅ cଷvଷ ൌ 0 The augmented matrix takes the form ൥ 1 0 െ2 2 1 0 3 2 1 อ 0 0 0 ൩ ሾvଵ vଶ vଷ | vሿ
Linear Dependence and Independence Page | 24 By reducing it to the reduced row echelon form, we obtain ൥ 1 0 െ2 2 1 0 3 2 1 อ 0 0 0 ൩ ିଶோభାோమ→ோమ ିଷோభାோయ→ோయ ሱ⎯⎯⎯⎯⎯⎯⎯⎯ሮ ൥ 1 0 െ2 0 1 4 0 2 7 อ 0 0 0 ൩ ିଶோమାோయ→ோయ ሱ⎯⎯⎯⎯⎯⎯⎯⎯ሮ ൥ 1 0 െ2 0 1 4 0 0 െ1 อ 0 0 0 ൩ ିோయ↔ோయ ሱ⎯⎯⎯⎯ሮ ൥ 1 0 െ2 0 1 4 0 0 1 อ 0 0 0 ൩ ିସோయାோమ→ோమ ଶோయାோభ→ோభ ሱ⎯⎯⎯⎯⎯⎯⎯⎯ሮ ൥ 100 010 001 อ 0 0 0 ൩ This shows the system has the only trivial solution cଵ ൌ cଶ ൌ cଷ ൌ 0. Therefore, S is linearly independent. Example 10.2.8. Check if Sൌ൝൥ 1 2 3 ൩, ൥ 0 1 2 ൩, ൥ െ1 0 1 ൩ൡ is linearly independent. Solution. To solve the following vector equation cଵvଵ ൅ cଶvଶ ൅ cଷvଷ ൌ 0, we apply the same procedure as in Example 10.2.7 to the following matrix. By reducing it to the reduced row echelon form, we obtain ൥ 1 0 െ1 2 1 0 3 2 1 อ 0 0 0 ൩ ିଶோభାோమ→ோమ ିଷோభାோయ→ோయ ሱ⎯⎯⎯⎯⎯⎯⎯⎯ሮ ൥ 1 0 െ1 0 1 2 0 2 4 อ 0 0 0 ൩ ିଶோమାோయ→ோయ ሱ⎯⎯⎯⎯⎯⎯⎯⎯ሮ ൥ 1 0 െ1 0 1 2 0 0 0 อ 0 0 0 ൩ This shows that cଷ is free variable, hence the system has a nontrivial solution. cଵ െ cଷ ൌ 0 cଶ ൅ 2cଷ ൌ 0 For example, if cଷ ൌ 1, then cଵ ൌ 1 and cଶ ൌ െ2. That is vଵ ൅ ሺെ2ሻvଶ ൅ vଷ ൌ 0 ൥ 1 2 3 ൩ െ 2 ൥ 0 1 2 ൩ ൅ ൥ െ1 0 1 ൩ ൌ ൥ 0 0 0 ൩ Hence S is linearly dependent.
Linear Dependence and Independence Page | 25 Remark If S ൌ ሼvଵ, vଶ,..., v௞, 0ሽ, then S is linearly dependent. In other words, if a set contains the zero vector, it must be linearly dependent. Because the vector equation cଵvଵ ൅ cଶvଶ ൅⋯൅c௞v௞ ൅ c௞ାଵ0ൌ0. has nontrivial solution, namely 0vଵ ൅ 0vଶ ൅⋯൅ 0v௞ ൅ ሺ1ሻ0ൌ0. Theorem 10.2.2. A set S ൌ ሼvଵ, vଶ,..., v௜,..., v௡ሽ is linearly dependent if and only if at least one of the vectors in S is a linear combination of the other vectors in S. Proof. See Appendix. Corollary A set S ൌ ሼvଵ, vଶሽ of two nonzero vectors in V is linearly dependent if and only if vଶ ൌ cvଵ for some scalar c.