Math 313 - Linear Algebra § 1.7 - 1.10 September 9 - 16, 2016 There is no royal road to geometry. - Euclid to Ptolemy
§ 1.7 Linear Independence Definition (Linear Independence) An indexed set of vectors { v 1 , . . . , v p } in R n is said to be linearly independent if the vector equation x 1 v 1 + x 2 v 2 + . . . + x p v p = 0 has only the trivial solution. The set { v 1 , . . . , v p } is said to be linearly dependent if there exist weights c 1 , . . . , c p , not all zero such that c 1 v 1 + c 2 v 2 + . . . + c p v p = 0 (1) In this case equation (1) is called a dependence relation .
§ 1.7 Linear Independence Example: Determine if the vectors 1 2 3 , , v 1 = 4 v 2 = 5 and v 3 = 6 7 8 9 are linearly independent. Example: Determine if 1 1 1 , , w 1 = 0 w 2 = 0 and w 3 = 1 1 2 0 are linearly independent.
§ 1.7 Linear Independence Saying that the vector equation x 1 v 1 + x 2 v 2 + . . . + x p v p = 0 has only the trivial solution x 1 = · · · = x p = 0 is the same as saying that 0 x 1 x 2 0 � � · · · v 1 v 2 v p = . . . . . . x p 0 has only the trivial solution x = 0 . Fact The columns of a matrix A are linearly independent if and only if the equation A x = 0 has only the trivial solution.
§ 1.7 Linear Independence Fact 1. A set { v } containing a single vector is linearly independent if and only if v � = 0 . 2. A set { v , w } is linearly independent if and only if neither of v not w is a scalar multiple of the other. Warning! A set of 3 or more vectors may be linearly dependent even though none of them is a scalar multiple of another vector in the set.
§ 1.7 Linear Independence Example: Determine if 2 16 , w 1 = 1 and w 2 = 8 − 1 − 7 are linearly independent.
§ 1.7 Linear Independence Theorem An indexed set S = { v 1 , . . . , v p } of two or more vectors is linearly dependent if and only if at least one of the vectors in S is a linear combination of the others. In fact, if S is linearly dependent and v 1 � = 0 , then some v j (with j > 1) is a linear combination of the preceding vectors, v 1 , . . . , v j − 1 . Warning! This theorem does not say that every vector of a linearly independent set can be written as a linear combination of the other vectors, just that some vector can.
§ 1.7 Linear Independence 1 1 and v = Example: Consider u = 2 1 . What is 0 0 Span { u , v } ? If w is another vector in R 3 , where will w lie if { u , v , w } is linearly independent? Where will it lie if { u , v , w } is linearly dependent?
§ 1.7 Linear Independence Theorem If a set contains more vectors than there are entries in each vector, then the set is linearly dependent. That is, any set { v 1 , . . . , v p } in R n is linearly dependent if p > n . Theorem If a set S = { v 1 , . . . , v p } in R n contains the zero vector, then the set is linearly dependent.
§ 1.7 Linear Independence Example: Which of the following sets of vectors is linearly independent? 1 1 − 3 5 , , , 1. 2 2 3 5 . 0 1 − 2 2 1 0 − 3 , , 2. 2 0 3 . 0 0 − 2 2 3 − 3 2 4 − 3 3. , , 8 5 − 12 4 6 − 6
§ 1.8 Introduction to Linear Transformations Given an m × n matrix A and a vector x ∈ R n , we can multiply A and x to give a vector in R m . We can think of an m × n matrix as taking vectors in R n and transforming them to vectors in R m . � 1 2 � − 2 3 ∈ R 3 . Then Example: If A = and x = − 2 1 1 7 5 � 1 � 21 � 2 � − 2 3 = ∈ R 2 . A x = − 2 1 1 7 35 5
§ 1.8 Introduction to Linear Transformations A transformation (or function or mapping ) T from R n to R m is a rule that assigns to each vector x in R n a vector T ( x ) in R m . The set R n is called the domain of T , and the set R m is called the codomain of T . Notation: T : R n → R m x �→ T ( x ) For a vector x ∈ R n , the vector T ( x ) in R m is called the image of x , and the set of all images T ( x ) is called the range of T .
§ 1.8 Introduction to Linear Transformations T x T ( x ) range codomain = R m domain = R n
§ 1.8 Introduction to Linear Transformations Example: Let T ( x ) = A x for all x ∈ R 3 , where � 1 � − 2 3 A = . 1 1 7 What is the domain and codomain of T ? Let � − 4 � − 2 1 � � . b = c = and u = 7 , − 5 − 5 − 4 Compute T ( u ). Are b and c in the range of T ? If so, find vectors x and v with T ( x ) = b and T ( v ) = c . What is the range of T ?
§ 1.8 Introduction to Linear Transformations Definition A transformation T is linear if for all u , v in the domain of T and all scalars c ∈ R 1. T ( u + v ) = T ( u ) + T ( v ) and 2. T ( c u ) = cT ( u ). Fact If T is a linear transformation, then for all vectors u , v in the domain of T , and all scalars c, d T ( 0 ) = 0 , and T ( c u + d v ) = cT ( u ) + dT ( v ) .
§ 1.8 Introduction to Linear Transformations Example: Let � 1 1 0 0 2 � , I = 0 1 0 B = , 0 1 0 0 1 � 0 1 0 0 � − 1 , C = 0 1 0 D = , 1 0 0 0 0 � 1 1 0 0 � . E = and F = 0 1 , 0 − 2 0 0 Described the linear transformations defined by these matrices. The matrix I above is called the identity matrix .
§ 1.9 The Matrix of a Linear Transformation Recall that for any n , we can define the following vectors in R n : 1 0 0 0 0 1 0 0 0 0 1 0 e 1 = e 2 = e 3 = e n = , , , . . . , . . . . . . . . . . . . 0 0 0 1
§ 1.9 The Matrix of a Linear Transformation Theorem Let T : R n → R m be a linear transformation. Then there exists a unique matrix A such that T ( x ) = A x for all x in R n . In fact, A is the m × n matrix whose j th column is the vector T ( e j ) where e j is the j th column of the identity matrix I n in R n : A = [ T ( e 1 ) T ( e n )] . . . . The matrix given above is called the standard matrix for the linear transformation T .
§ 1.9 The Matrix of a Linear Transformation Example: Find the standard matrix of the linear transformation T : R 2 → R 2 which reflects vectors in the line y = − x . Example: Find the standard matrix of the transformation S : R 3 → R 3 which reflects every vector through the xy -plane, and then projects to the xz -plane.
§ 1.9 The Matrix of a Linear Transformation Definition A transformation T : R n → R m is called onto if every vector b ∈ R m is the image of at least one vector in R n . T S x x T ( x ) S ( x ) range codomain = range = R m domain = R n domain = R n codomain = R m T is not onto. S is onto.
§ 1.9 The Matrix of a Linear Transformation Definition A transformation T : R n → R m is called one-to-one if every vector b ∈ R m is the image of at most one vector in R n . T S 0 0 0 0 range range domain = R n codomain = R m domain = R n codomain = R m T is not one-to-one. S is one-to-one.
§ 1.9 The Matrix of a Linear Transformation Example: Let T : R 4 → R 3 be a linear transformation with standard matrix 2 1 8 1 . A = 0 2 1 0 0 0 0 − 3 Is T one-to-one? Is T onto?
§ 1.9 The Matrix of a Linear Transformation Theorem Let T : R n → R m be a linear transformation, with standard matrix A (i.e. T ( x ) = A x for all vectors x ∈ R n ). Then the following are equivalent: 1. T is onto, 2. the columns of A span R m , 3. A has a pivot in every row. Proof. This is essentially Theorem 4 in § 1.4.
§ 1.9 The Matrix of a Linear Transformation Theorem Let T : R n → R m be a linear transformation, with standard matrix A (i.e. T ( x ) = A x for all vectors x ∈ R n ). Then the following are equivalent: 1. T is one-to-one, 2. T ( x ) = 0 has only the trivial solution x = 0 , 3. the columns of A are linearly independent, 4. A has a pivot in every column. Proof. We prove 1 ⇒ 4 ⇒ 2 ⇒ 3 ⇒ 1.
§ 1.10 Linear Models in Business, Science and Engineering Difference Equations: Consider an influenza strain, which has the following properties. Each week, an uninfected person has a 1% chance of catching the disease. Each week, an infected person has a 70% percent chance of recovering, and a 30% percent chance of remaining sick. Suppose that at week zero, 1000 people in a population of 100 000 are infected. Find a matrix equation to model this situation. � # of healthy at week k � � 999 000 � Let x k = , then x 0 = . # of sick at week k 1000
§ 1.10 Linear Models in Business, Science and Engineering Let � 0 . 99 0 . 7 � A = . 0 . 01 0 . 3 Then for all k ≥ 1, x k = A x k − 1 . � 989710 � � 987016 � x 1 = A x 0 = , x 2 = A x 1 = , 10290 12984 � 986235 � � 986008 � x 3 = A x 2 = , and x 4 = A x 3 = . 13765 13992
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