MATH 116 - WEEK 8 Sibel Özkan 1 Gebze Technical University April 12, 2022 1 (MATH 116 - LINEAR ALGEBRA) 12/04/2022 1 / 26 Today Span of a Set of Vectors (MATH 116 - LINEAR ALGEBRA) 12/04/2022 2 / 26 Today Span of a Set of Vectors Linear Independence (MATH 116 - LINEAR ALGEBRA) 12/04/2022 2 / 26 Today Span of a Set of Vectors Linear Independence Basis and Dimension (MATH 116 - LINEAR ALGEBRA) 12/04/2022 2 / 26 Span of a Set of Vectors Definition: (MATH 116 - LINEAR ALGEBRA) 12/04/2022 3 / 26 Example Example: Given the set of vectors −1 −1 1 u = 11 , v = 3 , w = 1 2 2 −2 Is the vector u is in the Span {v, w} ? (MATH 116 - LINEAR ALGEBRA) 12/04/2022 4 / 26 Example Example: Given the set of vectors −1 −1 1 u = 11 , v = 3 , w = 1 2 2 −2 Is the vector u is in the Span {v, w} ? Solution: Since −1 −1 1 11 = 3 3 + 2 1 2 2 −2 (MATH 116 - LINEAR ALGEBRA) 12/04/2022 4 / 26 Example Example: Given the set of vectors −1 −1 1 u = 11 , v = 3 , w = 1 2 2 −2 Is the vector u is in the Span {v, w} ? Solution: Since −1 −1 1 11 = 3 3 + 2 1 2 2 −2 Yes. (MATH 116 - LINEAR ALGEBRA) 12/04/2022 4 / 26 Example Example: In R3 , the span of e1 and e2 is the set of vectors of the form (MATH 116 - LINEAR ALGEBRA) 12/04/2022 5 / 26 Example Example: In R3 , the span of e1 and e2 is the set of vectors of the form α αe1 + βe2 = β 0 (MATH 116 - LINEAR ALGEBRA) 12/04/2022 5 / 26 Span of a Set of Vectors As an exercise, verify that Span(e1 , e2 ) is a subspace of R3 (MATH 116 - LINEAR ALGEBRA) 12/04/2022 6 / 26 Span of a Set of Vectors As an exercise, verify that Span(e1 , e2 ) is a subspace of R3 (MATH 116 - LINEAR ALGEBRA) 12/04/2022 6 / 26 Span of a Set of Vectors As an exercise, verify that Span(e1 , e2 ) is a subspace of R3 Also the span of e1 , e2 , e3 is the set of all vectors of the form (MATH 116 - LINEAR ALGEBRA) 12/04/2022 6 / 26 Span of a Set of Vectors As an exercise, verify that Span(e1 , e2 ) is a subspace of R3 Also the span of e1 , e2 , e3 is the set of all vectors of the form α1 α 1 e1 + α 2 e2 + α 3 e3 = α 2 α3 (MATH 116 - LINEAR ALGEBRA) 12/04/2022 6 / 26 Span of a Set of Vectors As an exercise, verify that Span(e1 , e2 ) is a subspace of R3 Also the span of e1 , e2 , e3 is the set of all vectors of the form α1 α 1 e1 + α 2 e2 + α 3 e3 = α 2 α3 So it is all of R3 . (MATH 116 - LINEAR ALGEBRA) 12/04/2022 6 / 26 Span of a Set of Vectors Theorem If v1 , v2 , . . . , vn are elements of a vector space V , then Span(v1 , v2 , . . . , vn ) is a subspace of V . (MATH 116 - LINEAR ALGEBRA) 12/04/2022 7 / 26 Span of a Set of Vectors Theorem If v1 , v2 , . . . , vn are elements of a vector space V , then Span(v1 , v2 , . . . , vn ) is a subspace of V . (MATH 116 - LINEAR ALGEBRA) 12/04/2022 7 / 26 Linear Independence Think of the following vectors in R3 1 −2 −1 x1 = −1 , x2 = 3 , x3 = 3 2 1 8 (MATH 116 - LINEAR ALGEBRA) 12/04/2022 8 / 26 Linear Independence Think of the following vectors in R3 1 −2 −1 x1 = −1 , x2 = 3 , x3 = 3 2 1 8 Let S be the subspace of R3 spanned by x1 , x2 , x3 . (MATH 116 - LINEAR ALGEBRA) 12/04/2022 8 / 26 Linear Independence Think of the following vectors in R3 1 −2 −1 x1 = −1 , x2 = 3 , x3 = 3 2 1 8 Let S be the subspace of R3 spanned by x1 , x2 , x3 . S can be represented in terms of only x1 and x2 since x3 is already in the span of x1 and x2 : x3 = 3x1 + 2x2 (MATH 116 - LINEAR ALGEBRA) 12/04/2022 8 / 26 Linear Independence Think of the following vectors in R3 1 −2 −1 x1 = −1 , x2 = 3 , x3 = 3 2 1 8 Let S be the subspace of R3 spanned by x1 , x2 , x3 . S can be represented in terms of only x1 and x2 since x3 is already in the span of x1 and x2 : x3 = 3x1 + 2x2 So, any linear combination of x1 , x2 and x3 cab be reduced to the linear combination of x1 and x2 : α1 x1 + α2 x2 + α3 x3 = α1 x1 + α2 x2 + α3 (3x1 + 2x2 ) = (α1 + 3α3 )x1 + (α2 + α3 )x2 (MATH 116 - LINEAR ALGEBRA) 12/04/2022 8 / 26 Linear Independence Think of the following vectors in R3 1 −2 −1 x1 = −1 , x2 = 3 , x3 = 3 2 1 8 Let S be the subspace of R3 spanned by x1 , x2 , x3 . S can be represented in terms of only x1 and x2 since x3 is already in the span of x1 and x2 : x3 = 3x1 + 2x2 So, any linear combination of x1 , x2 and x3 cab be reduced to the linear combination of x1 and x2 : α1 x1 + α2 x2 + α3 x3 = α1 x1 + α2 x2 + α3 (3x1 + 2x2 ) = (α1 + 3α3 )x1 + (α2 + α3 )x2 So, S= Span(x1 , x2 , x3 ) = Span(x1 , x2 ) (MATH 116 - LINEAR ALGEBRA) 12/04/2022 8 / 26 Linear Independence The above equation can be rewritten as 3x1 + 2x2 − 1x3 = 0 (MATH 116 - LINEAR ALGEBRA) 12/04/2022 9 / 26 Linear Independence The above equation can be rewritten as 3x1 + 2x2 − 1x3 = 0 Since the coefficients are not zero, we can solve for any vector in terms of the other two: x1 = (−2/3)x2 + (1/3)x3 , x2 = (−3/2)x1 + (1/2)x3 , x3 = 3x1 + 2x2 (MATH 116 - LINEAR ALGEBRA) 12/04/2022 9 / 26 Linear Independence The above equation can be rewritten as 3x1 + 2x2 − 1x3 = 0 Since the coefficients are not zero, we can solve for any vector in terms of the other two: x1 = (−2/3)x2 + (1/3)x3 , x2 = (−3/2)x1 + (1/2)x3 , x3 = 3x1 + 2x2 So, Span(x1 , x2 , x3 ) = Span(x2 , x3 ) = Span(x1 , x3 ) = Span(x1 , x2 ) (MATH 116 - LINEAR ALGEBRA) 12/04/2022 9 / 26 Linear Independence The above equation can be rewritten as 3x1 + 2x2 − 1x3 = 0 Since the coefficients are not zero, we can solve for any vector in terms of the other two: x1 = (−2/3)x2 + (1/3)x3 , x2 = (−3/2)x1 + (1/2)x3 , x3 = 3x1 + 2x2 So, Span(x1 , x2 , x3 ) = Span(x2 , x3 ) = Span(x1 , x3 ) = Span(x1 , x2 ) On the otherhand, there is no such dependency relation between x1 and x2 . (MATH 116 - LINEAR ALGEBRA) 12/04/2022 9 / 26 Linear Independence The above equation can be rewritten as 3x1 + 2x2 − 1x3 = 0 Since the coefficients are not zero, we can solve for any vector in terms of the other two: x1 = (−2/3)x2 + (1/3)x3 , x2 = (−3/2)x1 + (1/2)x3 , x3 = 3x1 + 2x2 So, Span(x1 , x2 , x3 ) = Span(x2 , x3 ) = Span(x1 , x3 ) = Span(x1 , x2 ) On the otherhand, there is no such dependency relation between x1 and x2 . So, if c1 x1 + c2 x2 = 0, only solution is c1 = c2 = 0 (MATH 116 - LINEAR ALGEBRA) 12/04/2022 9 / 26 Linear Independence We can generalize these observations as: (MATH 116 - LINEAR ALGEBRA) 12/04/2022 10 / 26 Linear Independence Definition: (MATH 116 - LINEAR ALGEBRA) 12/04/2022 11 / 26 Linear Independence Definition: 1 1 Example: The vectors and are linearly independent. 1 2 (MATH 116 - LINEAR ALGEBRA) 12/04/2022 11 / 26 Linear Independence Definition: 1 1 Example: The vectors and are linearly independent. 1 2 Since if 1 1 0 c1 + c2 = 1 2 0 (MATH 116 - LINEAR ALGEBRA) 12/04/2022 11 / 26 Linear Independence Definition: 1 1 Example: The vectors and are linearly independent. 1 2 Since if 1 1 0 c1 + c2 = 1 2 0 then c1 + c2 = 0, c1 + 2c2 = 0 and only solution to the system is c1 = c2 = 0 (MATH 116 - LINEAR ALGEBRA) 12/04/2022 11 / 26 Linear Dependence Definition: (MATH 116 - LINEAR ALGEBRA) 12/04/2022 12 / 26 Linear Dependence Definition: Example: Let x = (1, 2, 3)T . The vectors e1 , e2 , e3 and x are linearly dependent since (MATH 116 - LINEAR ALGEBRA) 12/04/2022 12 / 26 Linear Dependence Definition: Example: Let x = (1, 2, 3)T . The vectors e1 , e2 , e3 and x are linearly dependent since e1 + 2e2 + 3e3 − x = 0 (MATH 116 - LINEAR ALGEBRA) 12/04/2022 12 / 26 Linear Dependence Definition: Example: Let x = (1, 2, 3)T . The vectors e1 , e2 , e3 and x are linearly dependent since e1 + 2e2 + 3e3 − x = 0 (See in this case c1 = 1, c2 = 2, c3 = 3, c4 = −1) (MATH 116 - LINEAR ALGEBRA) 12/04/2022 12 / 26 Geometric Interpretation (MATH 116 - LINEAR ALGEBRA) 12/04/2022 13 / 26 Geometric Interpretation (MATH 116 - LINEAR ALGEBRA) 12/04/2022 13 / 26 Example Example: Is the set (1, 1, 1)T , (1, 1, 0)T , (1, 0, 0)T linearly independent? (MATH 116 - LINEAR ALGEBRA) 12/04/2022 14 / 26 Example Example: Is the set (1, 1, 1)T , (1, 1, 0)T , (1, 0, 0)T linearly independent? Solution: We must show that all c are zeros below. c1 (1, 1, 1)T + c2 (1, 1, 0)T + c3 (1, 0, 0)T = (0, 0, 0)T (MATH 116 - LINEAR ALGEBRA) 12/04/2022 14 / 26 Example Example: Is the set (1, 1, 1)T , (1, 1, 0)T , (1, 0, 0)T linearly independent? Solution: We must show that all c are zeros below. c1 (1, 1, 1)T + c2 (1, 1, 0)T + c3 (1, 0, 0)T = (0, 0, 0)T We have (MATH 116 - LINEAR ALGEBRA) 12/04/2022 14 / 26 Example Example: Is the set (1, 1, 1)T , (1, 1, 0)T , (1, 0, 0)T linearly independent? Solution: We must show that all c are zeros below. c1 (1, 1, 1)T + c2 (1, 1, 0)T + c3 (1, 0, 0)T = (0, 0, 0)T We have which gives us c1 = 0, c2 = 0, c3 = 0. Answer is YES. (MATH 116 - LINEAR ALGEBRA) 12/04/2022 14 / 26 Example Example: Is the set (1, 0, 1)T , (0, 1, 0)T linearly independent? (MATH 116 - LINEAR ALGEBRA) 12/04/2022 15 / 26 Example Example: Is the set (1, 0, 1)T , (0, 1, 0)T linearly independent? Solution: We must show that all c’s are zeros c1 (1, 0, 1)T + c2 (0, 1, 0)T = (0, 0, 0)T (MATH 116 - LINEAR ALGEBRA) 12/04/2022 15 / 26 Example Example: Is the set (1, 0, 1)T , (0, 1, 0)T linearly independent? Solution: We must show that all c’s are zeros c1 (1, 0, 1)T + c2 (0, 1, 0)T = (0, 0, 0)T which is (c1 , c2 , c1 )T = (0, 0, 0)T (MATH 116 - LINEAR ALGEBRA) 12/04/2022 15 / 26 Example Example: Is the set (1, 0, 1)T , (0, 1, 0)T linearly independent? Solution: We must show that all c’s are zeros c1 (1, 0, 1)T + c2 (0, 1, 0)T = (0, 0, 0)T which is (c1 , c2 , c1 )T = (0, 0, 0)T The answer is YES. (MATH 116 - LINEAR ALGEBRA) 12/04/2022 15 / 26 Theorem Theorem Let x1 , x2 , . . . , xn be n vectors in Rn and let X = (x1 x2 . . . xn ). The vectors x1 , x2 , . . . , xn are linearly dependent if and only if X is singular. (MATH 116 - LINEAR ALGEBRA) 12/04/2022 16 / 26 Theorem Theorem Let x1 , x2 , . . . , xn be n vectors in Rn and let X = (x1 x2 . . . xn ). The vectors x1 , x2 , . . . , xn are linearly dependent if and only if X is singular. Proof: The equation c1 x1 + c2 x2 + · · · + cn xn = 0 can be written as the matrix equation (MATH 116 - LINEAR ALGEBRA) 12/04/2022 16 / 26 Theorem Theorem Let x1 , x2 , . . . , xn be n vectors in Rn and let X = (x1 x2 . . . xn ). The vectors x1 , x2 , . . . , xn are linearly dependent if and only if X is singular. Proof: The equation c1 x1 + c2 x2 + · · · + cn xn = 0 can be written as the matrix equation Xc = 0 (MATH 116 - LINEAR ALGEBRA) 12/04/2022 16 / 26 Theorem Theorem Let x1 , x2 , . . . , xn be n vectors in Rn and let X = (x1 x2 . . . xn ). The vectors x1 , x2 , . . . , xn are linearly dependent if and only if X is singular. Proof: The equation c1 x1 + c2 x2 + · · · + cn xn = 0 can be written as the matrix equation Xc = 0 This equation has non-trivial (non-zero) solutions if and only if X is singular. (MATH 116 - LINEAR ALGEBRA) 12/04/2022 16 / 26 Example Example: Is the set (4, 2, 3)T , (2, 3, 1)T , (2, −5, 3)T linearly independent? (MATH 116 - LINEAR ALGEBRA) 12/04/2022 17 / 26 Example Example: Is the set (4, 2, 3)T , (2, 3, 1)T , (2, −5, 3)T linearly independent? Solution: Since 4 2 2 2 3 −5 = 0 3 1 3 (MATH 116 - LINEAR ALGEBRA) 12/04/2022 17 / 26 Example Example: Is the set (4, 2, 3)T , (2, 3, 1)T , (2, −5, 3)T linearly independent? Solution: Since 4 2 2 2 3 −5 = 0 3 1 3 NO: The vectors are linearly dependent. (MATH 116 - LINEAR ALGEBRA) 12/04/2022 17 / 26 Example : Example: Is the set (1, 1, 1)T , (1, 1, 0)T , (1, 0, 0)T linearly independent? (MATH 116 - LINEAR ALGEBRA) 12/04/2022 18 / 26 Example : Example: Is the set (1, 1, 1)T , (1, 1, 0)T , (1, 0, 0)T linearly independent? Solution: Since 1 1 1 0 1 1 =1 0 0 1 (MATH 116 - LINEAR ALGEBRA) 12/04/2022 18 / 26 Example : Example: Is the set (1, 1, 1)T , (1, 1, 0)T , (1, 0, 0)T linearly independent? Solution: Since 1 1 1 0 1 1 =1 0 0 1 YES. The vectors are linearly independent. (MATH 116 - LINEAR ALGEBRA) 12/04/2022 18 / 26 Example : Example: Is the set (1, −1, 2, 3)T , (−2, 3, 1, −2)T , (1, 0, 7, 7)T linearly independent? (MATH 116 - LINEAR ALGEBRA) 12/04/2022 19 / 26 Example : Example: Is the set (1, −1, 2, 3)T , (−2, 3, 1, −2)T , (1, 0, 7, 7)T linearly independent? Solution: Let’s put them in Augmented Matrix form (MATH 116 - LINEAR ALGEBRA) 12/04/2022 19 / 26 Example : Example: Is the set (1, −1, 2, 3)T , (−2, 3, 1, −2)T , (1, 0, 7, 7)T linearly independent? Solution: Let’s put them in Augmented Matrix form (MATH 116 - LINEAR ALGEBRA) 12/04/2022 19 / 26 Example : Example: Is the set (1, −1, 2, 3)T , (−2, 3, 1, −2)T , (1, 0, 7, 7)T linearly independent? Solution: Let’s put them in Augmented Matrix form Since the REF involves a free variable c3 , there are nontrivial solutions, hence the vectors are linearly dependent. (MATH 116 - LINEAR ALGEBRA) 12/04/2022 19 / 26 Theorem Theorem Let v1 , v2 , . . . , vn be vectors in a vector space V . A vector v ∈ Span (v1 , v2 , . . . , vn ) can be written uniquely as a linear combination of v1 , v2 , . . . , vn if and only if v1 , v2 , . . . , vn are linearly independent. (MATH 116 - LINEAR ALGEBRA) 12/04/2022 20 / 26 Basis and Dimension Definition The vectors v1 , v2 , . . . , vn form a basis for a vector space V if and only if (i) v1 , v2 , . . . , vn are linearly independent. (ii) v1 , v2 , . . . , vn span V . (MATH 116 - LINEAR ALGEBRA) 12/04/2022 21 / 26 Basis and Dimension Definition The vectors v1 , v2 , . . . , vn form a basis for a vector space V if and only if (i) v1 , v2 , . . . , vn are linearly independent. (ii) v1 , v2 , . . . , vn span V . Example: The standard basis for R3 is {e1 , e2 , e3 }; but there are many bases we could choose for R3 . Here are some examples: (MATH 116 - LINEAR ALGEBRA) 12/04/2022 21 / 26 Basis and Dimension Definition The vectors v1 , v2 , . . . , vn form a basis for a vector space V if and only if (i) v1 , v2 , . . . , vn are linearly independent. (ii) v1 , v2 , . . . , vn span V . Example: The standard basis for R3 is {e1 , e2 , e3 }; but there are many bases we could choose for R3 . Here are some examples: (MATH 116 - LINEAR ALGEBRA) 12/04/2022 21 / 26 Example (MATH 116 - LINEAR ALGEBRA) 12/04/2022 22 / 26 A Theorem and a Corollary Theorem If v1 , v2 , . . . , vn is a spanning set for a vector space V , then any collection of m vectors in V , where m > n, is linearly dependent. (MATH 116 - LINEAR ALGEBRA) 12/04/2022 23 / 26 A Theorem and a Corollary Theorem If v1 , v2 , . . . , vn is a spanning set for a vector space V , then any collection of m vectors in V , where m > n, is linearly dependent. Corollary If both v1 , v2 , . . . , vn and u1 , u2 , . . . , um are basis for a vector space V , then n = m. (MATH 116 - LINEAR ALGEBRA) 12/04/2022 23 / 26 Dimension (MATH 116 - LINEAR ALGEBRA) 12/04/2022 24 / 26 Dimension (MATH 116 - LINEAR ALGEBRA) 12/04/2022 24 / 26 Basis and Dimension Theorem If V is a vector space of dimension n > 0, then (i) any set of n linearly independent vectors spans V (ii) any n vectors that span V are linearly independent. (MATH 116 - LINEAR ALGEBRA) 12/04/2022 25 / 26 Basis and Dimension Theorem If V is a vector space of dimension n > 0, then (i) any set of n linearly independent vectors spans V (ii) any n vectors that span V are linearly independent. Example: Show that (1, 2, 3)T , (−2, 1, 0)T , (1, 0, 1)T is a basis for R3 . (MATH 116 - LINEAR ALGEBRA) 12/04/2022 25 / 26 Basis and Dimension Theorem If V is a vector space of dimension n > 0, then (i) any set of n linearly independent vectors spans V (ii) any n vectors that span V are linearly independent. Example: Show that (1, 2, 3)T , (−2, 1, 0)T , (1, 0, 1)T is a basis for R3 . Solution: Since dimR3 = 3, we only need to show that they are linearly independent. (MATH 116 - LINEAR ALGEBRA) 12/04/2022 25 / 26 Basis and Dimension Theorem If V is a vector space of dimension n > 0, then (i) any set of n linearly independent vectors spans V (ii) any n vectors that span V are linearly independent. Example: Show that (1, 2, 3)T , (−2, 1, 0)T , (1, 0, 1)T is a basis for R3 . Solution: Since dimR3 = 3, we only need to show that they are linearly independent. Since 1 −2 1 2 1 0 =2 3 0 1 (MATH 116 - LINEAR ALGEBRA) 12/04/2022 25 / 26 Basis and Dimension Theorem If V is a vector space of dimension n > 0, then (i) any set of n linearly independent vectors spans V (ii) any n vectors that span V are linearly independent. Example: Show that (1, 2, 3)T , (−2, 1, 0)T , (1, 0, 1)T is a basis for R3 . Solution: Since dimR3 = 3, we only need to show that they are linearly independent. Since 1 −2 1 2 1 0 =2 3 0 1 they are linearly independent and this set is a basis for R3 . (MATH 116 - LINEAR ALGEBRA) 12/04/2022 25 / 26 Basis and Dimension Theorem If V is a vector space of dimension n > 0, then (i) no set of fewer than n vectors can span V (ii) any subset of fewer than n linearly independent vectors can be extended to form a basis for V (iii) any spanning set containing more than n vectors can be pared down to form a basis for V . (MATH 116 - LINEAR ALGEBRA) 12/04/2022 26 / 26 David C. Lay, Linear Algebra and its Applications, Addison-Wesley (2012) S. J. Leon, Linear Algebra with Applications, Pearson (2014) (MATH 116 - LINEAR ALGEBRA) 12/04/2022 26 / 26