Lecture 6: Elif Tan Ankara University Elif Tan (Ankara University) Lecture 6 / 0
Definition (Linear Combination) Let v, v 2,..., v k be vectors in (V,, ) a vector space. A vector v V is called a linear combination of v, v 2,..., v k if for the scalars c, c 2,..., c k. v = c v c 2 v 2... c k v k, Elif Tan (Ankara University) Lecture 6 2 / 0
Definition (Spannig Set) Let S = {v, v 2,..., v k } be a set of vectors in a vector space V. Then the set of all vectors in V that are linear combinations of the vectors in S is denoted by SpanS, that is, SpanS = {c v c 2 v 2... c k v k c, c 2,..., c k R}. If every vector in V is a linear combination of the vectors in S, the set S is said to span V and denoted by SpanS = V. Theorem Let S = {v, v 2,..., v k } be a set of vectors in a vector space V. Then SpanS < V. Note that a vector space may have many spanning sets. Elif Tan (Ankara University) Lecture 6 3 / 0
To determine whether a vector v of V is in SpanS, we investigate the consistency of the corresponding linear system. Example Let V := R 3, consider the vectors 2 3 v = 2, v 2 =, v 3 = 3. Determine whether Span {v, v 2, v 3 } = V. Solution: Let v := a b c be an arbitrary vector in R 3. If we find the scalars c, c 2, and c 2 such that v = c v c 2 v 2 c 3 v 3, then the set {v, v 2, v 3 } spans R 3. Elif Tan (Ankara University) Lecture 6 4 / 0
The corresponding linear system is c + 2c 2 + 3c 3 = a 2c + c 2 3c 3 = b c c 2 + c 3 = c. If we transform the augmented matrix to the reduced row echelon form, we obtain c = (9a 5b 9c) /2 c 2 = (5a + 2b 9c) /2 c 3 = (a b + c) /7 which indicates the corresponding linear system is consistent. Thus, Span {v, v 2, v 3 } = V. Elif Tan (Ankara University) Lecture 6 5 / 0
Definition (Linear independency) The vectors v, v 2,..., v k in a vector space (V,, ) are said to be linearly independent if there exist scalars c, c 2,..., c k, not all zero, such that c v c 2 v 2... c k v k = 0. The vectors v, v 2,..., v k are called linealy independent if c = c 2 = = c k = 0 such that c v c 2 v 2... c k v k = 0. Elif Tan (Ankara University) Lecture 6 6 / 0
To determine whether a set of vectors is linearly independent or linarly dependent, we investigate the nontrivial (nonzero) solution of the corresponding homogenous linear system. If the system has a nontrivial solution, then the vectors are linearly dependent. If the system has only trivial (zero) solution, then the vectors are linearly independent. Elif Tan (Ankara University) Lecture 6 7 / 0
Example Determine whether the vectors v = are linearly independent. 2, v 2 = 2 Solution: The corresponding homogenous linear system is c + 2c 2 + 3c 3 = 0 2c + c 2 3c 3 = 0 c c 2 + c 3 = 0., v 3 = If we transform the augmented matrix to the reduced row echelon form, we obtain c = c 2 = c 3 = 0 which indicates the linear system has only zero solution. Thus the vectors are linearly independent. 3 3 Elif Tan (Ankara University) Lecture 6 8 / 0
Theorem Let S = {v, v 2,..., v n } be a set of vectors in a vector space R n. Let A be the matrix whose columns are the elements of S. Then S is linearly independent if and only if det (A) = 0. Example From the former example, since 2 3 A = 2 3 I 3, then det (A) = 0 which indicates the vectors are linearly independent. Elif Tan (Ankara University) Lecture 6 9 / 0
Theorem Let S and S 2 be finite subset of a vector space V and S S 2. Then (i) S is linearly dependent S 2 is linearly dependent (ii) S 2 is linearly independent S is linearly independent. Elif Tan (Ankara University) Lecture 6 0 / 0
Theorem Let S and S 2 be finite subset of a vector space V and S S 2. Then (i) S is linearly dependent S 2 is linearly dependent (ii) S 2 is linearly independent S is linearly independent. 2 If S is any set of vectors that contains 0, then S is linearly dependent. Elif Tan (Ankara University) Lecture 6 0 / 0
Theorem Let S and S 2 be finite subset of a vector space V and S S 2. Then (i) S is linearly dependent S 2 is linearly dependent (ii) S 2 is linearly independent S is linearly independent. 2 If S is any set of vectors that contains 0, then S is linearly dependent. 3 If S consists of a single nonzero vector, then S is linearly independent. Elif Tan (Ankara University) Lecture 6 0 / 0
Theorem Let S and S 2 be finite subset of a vector space V and S S 2. Then (i) S is linearly dependent S 2 is linearly dependent (ii) S 2 is linearly independent S is linearly independent. 2 If S is any set of vectors that contains 0, then S is linearly dependent. 3 If S consists of a single nonzero vector, then S is linearly independent. 4 If S = {v, v 2,..., v k } is linearly independent, then the vectors v, v 2,..., v k must be distinct and nonzero. Elif Tan (Ankara University) Lecture 6 0 / 0
Theorem Let S and S 2 be finite subset of a vector space V and S S 2. Then (i) S is linearly dependent S 2 is linearly dependent (ii) S 2 is linearly independent S is linearly independent. 2 If S is any set of vectors that contains 0, then S is linearly dependent. 3 If S consists of a single nonzero vector, then S is linearly independent. 4 If S = {v, v 2,..., v k } is linearly independent, then the vectors v, v 2,..., v k must be distinct and nonzero. 5 Let S = {v, v 2,..., v k } spans a vector space V and v j is a linear combination of the preceding vectors in S. Then the set S {v j } also spans V. Elif Tan (Ankara University) Lecture 6 0 / 0