MTH 306 Spring Term 2007 Lesson 5 John Lee Oregon State University (Oregon State University) 1 / 12
Lesson 5 Goals: Be able to de ne linear independence and linear dependence of a set of vectors (Oregon State University) 2 / 12
Lesson 5 Goals: Be able to de ne linear independence and linear dependence of a set of vectors Interpret independence and dependence in geometric terms for vectors in 2- and 3-space (Oregon State University) 2 / 12
Lesson 5 Goals: Be able to de ne linear independence and linear dependence of a set of vectors Interpret independence and dependence in geometric terms for vectors in 2- and 3-space Use the determinant test or solve an appropriate linear system of equations to determine linear dependence or independence of a set of vectors (Oregon State University) 2 / 12
Lesson 5 Goals: Be able to de ne linear independence and linear dependence of a set of vectors Interpret independence and dependence in geometric terms for vectors in 2- and 3-space Use the determinant test or solve an appropriate linear system of equations to determine linear dependence or independence of a set of vectors Be able to nd a nontrivial linear combination among linearly dependent vectors (Oregon State University) 2 / 12
Redundant and Non-redundant Data Sets Our main goal is to determine when a set of vectors does or does not contain redundant information. (Oregon State University) 3 / 12
Redundant and Non-redundant Data Sets Our main goal is to determine when a set of vectors does or does not contain redundant information. In the former case, the vectors are called linearly dependent. (Oregon State University) 3 / 12
Redundant and Non-redundant Data Sets Our main goal is to determine when a set of vectors does or does not contain redundant information. In the former case, the vectors are called linearly dependent. In the latter case, the vectors are linearly independent. (Oregon State University) 3 / 12
Linear Combinations A nite sum of the form c 1 v 1 + c 2 v 2 + c 3 v 3 + + c k v k is called a linear combination of the vectors v 1, v 2, v 3,..., v k. (Oregon State University) 4 / 12
Linear Combinations A nite sum of the form c 1 v 1 + c 2 v 2 + c 3 v 3 + + c k v k is called a linear combination of the vectors v 1, v 2, v 3,..., v k. A linear combination is called nontrivial if at least one of the scalars is not zero. (Oregon State University) 4 / 12
Linear Combinations A nite sum of the form c 1 v 1 + c 2 v 2 + c 3 v 3 + + c k v k is called a linear combination of the vectors v 1, v 2, v 3,..., v k. A linear combination is called nontrivial if at least one of the scalars is not zero. The set of all linear combinations of v 1, v 2, v 3,..., v k is called the span of the vectors v 1, v 2, v 3,..., v k. (Oregon State University) 4 / 12
Example Describe 2 3the span 2 of3the following sets of vectors in R 3. 1 2 (a) 4 4 5 and 4 5 5 7 8 2 (b) 4 1 4 7 3 2 5, 4 2 5 8 3 2 5, and 4 5 11 17 What about redundancy in the two cases? 3 5 (Oregon State University) 5 / 12
Linear Dependence and Independence for Three Vectors in R 3 a, b, c are LD () a, b, c all lie in a plane in R 3. Why? (Oregon State University) 6 / 12
Linear Dependence and Independence for Three Vectors in R 3 a, b, c are LD () a, b, c all lie in a plane in R 3. a, b, c are LI () the span of a, b, and c is R 3. Why? (Oregon State University) 6 / 12
Linear Dependence A set of vectors v 1, v 2, v 3,..., v k is linearly dependent (LD) if there are scalars c 1, c 2,..., c k NOT ALL ZERO such that c 1 v 1 + c 2 v 2 + c 3 v 3 + + c k v k = 0. (Oregon State University) 7 / 12
Linear Dependence A set of vectors v 1, v 2, v 3,..., v k is linearly dependent (LD) if there are scalars c 1, c 2,..., c k NOT ALL ZERO such that c 1 v 1 + c 2 v 2 + c 3 v 3 + + c k v k = 0. If the vectors are in R n we may say the vectors are linearly dependent over R, emphasizing the scalars are real numbers. (Oregon State University) 7 / 12
Linear Dependence A set of vectors v 1, v 2, v 3,..., v k is linearly dependent (LD) if there are scalars c 1, c 2,..., c k NOT ALL ZERO such that c 1 v 1 + c 2 v 2 + c 3 v 3 + + c k v k = 0. If the vectors are in R n we may say the vectors are linearly dependent over R, emphasizing the scalars are real numbers. If the vectors are in C n we may say the vectors are linearly dependent over C, emphasizing the scalars are complex numbers. (Oregon State University) 7 / 12
Linear Dependence A set of vectors v 1, v 2, v 3,..., v k is linearly dependent (LD) if there are scalars c 1, c 2,..., c k NOT ALL ZERO such that c 1 v 1 + c 2 v 2 + c 3 v 3 + + c k v k = 0. If the vectors are in R n we may say the vectors are linearly dependent over R, emphasizing the scalars are real numbers. If the vectors are in C n we may say the vectors are linearly dependent over C, emphasizing the scalars are complex numbers. Corresponding language is used for linear independence. (Oregon State University) 7 / 12
Linear Independence A set of vectors v 1, v 2, v 3,..., v k is linearly independent (LI) if it is not linearly dependent. Consequently: A set of vectors v 1, v 2, v 3,..., v k is linearly independent if and only if the equation c 1 v 1 + c 2 v 2 + c 3 v 3 + + c k v k = 0 ) c 1 = 0, c 2 = 0, c 3 = 0,..., c k = 0. (Oregon State University) 8 / 12
Linear Combinations revisited Any linear combination of vectors in real or complex n-space can be expressed as Ac where (Oregon State University) 9 / 12
Linear Combinations revisited Any linear combination of vectors in real or complex n-space can be expressed as Ac where A is the matrix whose columns are the vectors in the linear combination (Oregon State University) 9 / 12
Linear Combinations revisited Any linear combination of vectors in real or complex n-space can be expressed as Ac where A is the matrix whose columns are the vectors in the linear combination and c is the column vector whose components are the corresponding scalar multiples of those vectors. (Oregon State University) 9 / 12
Linear Combinations revisited Any linear combination of vectors in real or complex n-space can be expressed as Ac where A is the matrix whose columns are the vectors in the linear combination and c is the column vector whose components are the corresponding scalar multiples of those vectors. In symbols c 1 v + c 2 w + = Ac where A = [v, w,...] and c = [c 1, c 2,...] and only a nite number of vectors and scalars are involved. (Oregon State University) 9 / 12
Tests for Linear Dependence and Linear Independence The equivalence c 1 v 1 + c 2 v 2 + + c k v k = 0 () Ac = 0 where A = [v 1, v 2,..., v k ] and c = [c 1, c 2,..., c k ] leads immediately to tests for LD and LI in terms of solving systems of linear equations: (Oregon State University) 10 / 12
Tests for Linear Dependence and Linear Independence The equivalence c 1 v 1 + c 2 v 2 + + c k v k = 0 () Ac = 0 where A = [v 1, v 2,..., v k ] and c = [c 1, c 2,..., c k ] leads immediately to tests for LD and LI in terms of solving systems of linear equations: v 1, v 2,..., v k are LD if and only if the homogeneous system Ac = 0 has nontrivial solutions for c. (Oregon State University) 10 / 12
Tests for Linear Dependence and Linear Independence The equivalence c 1 v 1 + c 2 v 2 + + c k v k = 0 () Ac = 0 where A = [v 1, v 2,..., v k ] and c = [c 1, c 2,..., c k ] leads immediately to tests for LD and LI in terms of solving systems of linear equations: v 1, v 2,..., v k are LD if and only if the homogeneous system Ac = 0 has nontrivial solutions for c. v 1, v 2,..., v k are LI if and only if the homogeneous system Ac = 0 has only the trivial solution c = 0. (Oregon State University) 10 / 12
Tests for Linear Dependence and Linear Independence The equivalence c 1 v 1 + c 2 v 2 + + c k v k = 0 () Ac = 0 where A = [v 1, v 2,..., v k ] and c = [c 1, c 2,..., c k ] leads immediately to tests for LD and LI in terms of solving systems of linear equations: v 1, v 2,..., v k are LD if and only if the homogeneous system Ac = 0 has nontrivial solutions for c. v 1, v 2,..., v k are LI if and only if the homogeneous system Ac = 0 has only the trivial solution c = 0. Consequently, a set of n, n-vectors is LI if and only if the determinant of the matrix whose columns are the given vectors is not zero. (Oregon State University) 10 / 12
Example Test the given set of vectors for LI or LD. If LD nd a nontrival linear combination of the vectors that has sum zero. 2 3 2 3 2 3 2 3 1 0 1 3 4 1 5, 4 1 5, 4 0 5, 4 2 5 0 1 1 1 (Oregon State University) 11 / 12
Example Test the given set of vectors for LI or LD. If LD nd a nontrival linear combination of the vectors that has sum zero. 2 6 6 4 1 1 0 1 3 7 7 5, 2 6 6 4 0 1 0 0 3 7 7 5, 2 6 6 4 1 0 1 2 3 7 7 5, 2 6 6 4 0 2 1 1 3 7 7 5 (Oregon State University) 12 / 12