MATH 2331 Linear Algebra Section 1.1 Systems of Linear Equations Finding the solution to a set of two equations in two variables: Example 1: Solve: x x = 3 1 2 2x + 4x = 12 1 2 Geometric meaning: Do these lines intersect? Solution: 1
In general, for a system of two equations in two variables, If the lines intersect at a point, then the system has one solution. If the lines are identical, then there are infinitely many solutions (any point on the line is a solution). If the lines are parallel, then the system does not have any solutions. 2
A linear equation in variables x1, x2,..., x n is an equation that can be written as: a1 x1 + a2x2 +... + anxn = b where b and the coefficients a1, a2,..., a n are real numbers usually known in advance. A system of linear equations (or a linear system) is a collection of one or more linear equations involving the same variables. For example: x + 5x x = 7 1 2 3 4x 2x + 7x = 5 1 2 3 is a linear system of two linear equations in 3 unknowns. Note: A linear system has Exactly one solution (a unique solution) No solution Or infinitely many solutions. The system is said to be consistent if it has one solution or infinitely many solutions, it is said to be inconsistent if it has no solution. 3
Example 1: Solve: x x = 3 1 2 2x + 4x = 12 1 2 How do you solve this system without graphing? 4
Matrix Notation: A matrix is a rectangular array of numbers. If it has m columns and n rows, then it is said to be a size m n matrix. Example 1: x x = 3 1 2 2x + 4x = 12 1 2 Coefficient Matrix: Augmented Matrix: 5
Example 2: x x + x = 4 1 2 3 x + 5x = 7 2 3 x + 2x + 4x = 3 1 2 3 Does this system have a solution? If so, what is the solution of this system? 6
Section 1.2: Row Reduction A rectangular matrix is in echelon form (or row echelon form) if it has the following three properties: 1) All nonzero rows are above any rows of all zeros. 2) Each leading entry of a row is in a column to the right of the leading entry of the row above it. 3) All entries in a column below a leading entry are zeros. If a matrix in echelon form satisfies the following additional conditions, then it is in reduced echelon form (or reduced row echelon form): 4) The leading entry in each nonzero row is 1. 5) Each leading 1 is the only nonzero entry in its column. 5 * * * * * 0 4 * * * * 0 0 0 2 * * 0 0 0 0 0 0 11 * * * * * * * 0 2 * * * * * * 0 0 0 5 * * * * 0 0 0 0 0 0 2 * 0 0 0 0 0 0 0 6 1 0 * 0 * * 0 1 * 0 * * 0 0 0 1 * * 0 0 0 0 0 0 1 0 * 0 * * 0 0 0 1 * 0 * * 0 0 0 0 0 1 * * 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 7
8 Example 3: Determine if the following matrices are in echelon form. a. 2 1 0 0 0 0 1 0 0 0 0 1 b. 2 0 1 0 3 0 0 1 0 0 0 0 c. 0 0 0 3 2 1 d. 0 0 0 0 3 1 0 0 0 0 9 1 e. 5 2 0 0 0 1 0 0 0 0 0 0 f. 1 0 10 5 0 2 10 2 0 0 0 0
Elementary row operations: We can find the echelon (or reduced echelon) form of any matrix by applying elementary row operations to that matrix. Elementary row operations include the following: 1) (Replacement) Replace one row by the sum of itself and a multiple of another row. 2) (Interchange) Interchange two rows. 3) (Scaling) Multiply all entries in a row by a nonzero constant. Two matrices are called row equivalent if there is a sequence of elementary row operations that transforms one matrix into the other. A pivot position in a matrix A is a location in A that corresponds to a leading 1 in the reduced form. A pivot column is a column of A that contains a pivot position. The variables corresponding to pivot columns are called basic variables, and the other variables are called free variables 9
Example 4: Find the echelon form of the matrix: 1 1 5 0 2 4 0 1 1 3 1 2 10
To solve a linear system: Write the augmented matrix representing the system; [ A b ] and perform the necessary row operations on [ A b ] to find the echelon or reduced echelon form of this matrix and rewrite the system. Example 5: Solve the system using row reduction: 2x + x 2x = 3 1 2 3 4x + x + x = 15 1 2 3 6x + 2x + x = 14 1 2 3 11
Example 6: Solve the system using row reduction: x 2x = 1 2 3 4x + x + x = 6 1 2 3 2x + x 2x = 1 1 2 3 12
Example 7: Solve the system using row reduction: 2x + x 2x = 1 x 1 2 3 + 2x = 5 2 3 6x + 3x 6x = 3 1 2 3 13
Example 8: Solve the system using row reduction: 2x + x 2x = 1 x 1 2 3 + 2x = 5 2 3 6x + 3x 6x = 7 1 2 3 14
Theorem 2: Existence and uniqueness A linear system is consistent if and only if the augmented matrix does not have any rows of the form: [ 0 0 b] where b is nonzero. If a linear system is consistent, then it has a unique solution if there are no free variables it has infinitely many solutions if there is at least one free variable. Example 9: Given the following augmented matrices in row-echelon form, what are the solutions? 1 2 3 4 0 0 0 2 1 0 2 4 0 5 0 4 8 0 8 24 0 0 0 0 0 12 1 0 0 4 0 1 0 5 0 0 1 6 1 2 3 4 0 0 0 0 1 2 3 4 0 0 1 2 15
Example 10: Find the general solution of the linear system whose augmented matrix has been reduced to: 1 5 0 0 4 0 0 0 1 0 6 10 0 0 0 1 2 8 16
(Extra) Example 11: Find the general solution of the linear system whose augmented matrix has been reduced to: 1 0 2 4 0 5 0 4 8 0 8 24 0 0 0 0 2 12 17
MATH 2331 Linear Algebra Section 1.3 Vector Equations A matrix with only one column is called a column vector, or simply a vector. For example: Notation: R 2 = the set of all vectors with two entries x 1 = : x1, x2 are real numbers x 2 Two vectors in 2 R are equal if the corresponding entries are equal. 1
Vector addition: =, = ; += + +. Scalar multiplication: =, = (the number c is called a scalar ). Example: Let 2 u = 1 and 0 v = 4. 2
Geometric Description of ℝ 2 : a We can identify a geometric point ( a, b) with the vector. b Scalar Multiplication 2 1 Example: Let u = ; Display 2u, u, u 2 1 3
Vector Addition 4
Example: Let 1 u = 1 and 1 v = 2. Display u + v, u v and 3u + 2v. 5
Notation: R 3 = the set of all vectors with 3 components x1 x = 2 : x1, x2, x3 are real numbers x 3 Vectors in 3 R are geometrically represented by points in a three-dimensional space. 6
Notation: R n = the set of all vectors with n components x1 x 2 = x 3 : xi are real numbers x n Note: The zero vector in n R is the vector whose all entries are 0. u v = u + ( v) Study the algebraic properties of n R in page 27. 7
Linear Combinations Definition: Given vectors v 1, v 2,..., v p in the vector y defined by y = c1v1 + c2v2 +... + cpvp n R and given scalars c 1, c 2,..., c p, is called a linear combination of v 1,, v p with weights c 1,, c p. The weights in a linear combination can be any real numbers, including zero. For vectors u and v ; cu + dv is a linear combination of u and v. For example: Give a linear combination of the vectors: 2 u = 4 and 1 v = 1. 8
Questions: If u and v are vectors in two-dimensional space; What are all combinations of u? What are all combinations of u and v? 1 2 1 Example: Give a linear combination of the vectors: =0, =5, = 2 2 1 2 Questions: If u, v, w are vectors in three-dimensional space; What are all combinations of u? What are all combinations of u and v? What are all combinations of u,v, and w? 9
Example: = 2 1, = 1 1 ; Can you find a linear combination of v and w that equals = 11 4? How about = 0 0? How about 9 b = 1. 5? 10
Vector Equations: A vector equation x1a1 + x2a2 +... + xnan = b has the same solution set as the linear system with augmented matrix: a1 a2... an b In particular, b can be written as a linear combination of a1, a2,..., a n if there exists a solution to the linear system corresponding to this augmented matrix. 11
Definition: If v1, v2,..., v p are vectors in R n, then { p} Span v, v,..., v = the set of all linear combinations of v, v,..., v 1 2 1 2 is called the subset of R n spanned (or generated) by v1, v2,..., v p. p { 1, 2,..., p} = { 1 1 + 2 2 +... + n n : i are scalars} Span v v v c v c v c v c Example: Let 1 u = 2 and 1 v = 0. Span{ u } = Span{ v } = Span{ u, v } = 12
Example: Let 1 u = 2 3 1 v = 0, 1 0 w = 1. 0 Span{ u } = Span{ u, v } = Span{ u, v, w } = 13
14
Example: Let 1 u = 2 3 and 1 v = 0. 1 Span{ u, v } is a plane through the origin in 3 R. Is 1 b = 6 in that plane? 5 15
Extra: Let 1 1 5 A = 2 4 0 1 3 1 Let W be the set of all linear combinations of the columns of A. Is b in W? 12 b = 6 0 1 b = 4 3 16
MATH 2331 Linear Algebra Section 1.4 The Matrix Equation Ax = b Let A be an m n matrix with columns a1, a2,..., a n and x be a vector in R n. The PRODUCT of A and x, denoted Ax, is the linear combination of the columns of A using the corresponding entries in x as weights. That is, x1 x Ax = a a a = x a + x a + + x a xn 2 1 2... n 1 1 2 2... n n Example: 1
Example: For u1, u2, u3 R m, write the linear combination 5u 2u + 4u as a matrix product. 1 2 3 2
Matrix Equations: An equation in the form Ax = b is called a matrix equation. Linear system: x + x + 2x = 6 1 2 3 x 4x = 7 2 3 Vector Equation: Matrix Equation: 3
Theorem: Let A be an m n matrix with columns a1, a2,..., a n and x be a vector in R n. The matrix equation equation Ax = b has the same solution set as the vector x a + x a + + x a = b 1 1 2 2... n n which has the same solution set as the system with the augmented matrix a1 a2... an b 4
Theorem: The matrix equation Ax = b has a solution if and only if the vector b is a linear combination of the columns of the matrix A. Example: 5
1 1 2 b1 Example: Let A 2 3 0 = and b = b2. 4 5 4 b 3 Is the equation Ax = b consistent for all possible b 1, b 2, b 3? 6
1 0 2 b1 Example: Let A 1 1 0 = and b = b2. 2 0 5 b 3 Is the equation Ax = b consistent for all possible b 1, b 2, b 3? 7
Theorem: Let A be an m n matrix. The following are equivalent: (i) The equation Ax = b has a solution for any b in R m. (ii) Each b in R m is a linear combination of the columns of A. (iii) The columns of A span R m. (iv) A has a pivot position in every row. 8
More about the Matrix-Vector Product Ax : Example: 1 2 3 A 4 5 6 = 7 8 9 and x x. Compute 1 = x2 x 3 Ax. 9
Row-vector rule: If Ax is defined, then row1 row1 x row2 row2 x Ax = x = row m row m x That is, the i th entry in Ax is row i times vector x. Example: 5 1 2 3 4 = 0 5 6 1 10
Example: 1 2 5 0 1 = 4 2 6 Example: 1 0 0 a 0 1 0 b = 0 0 1 c Homework: Study the properties of the Matrix-Vector product on Page 39 (Theorem 5). 11
MATH 2331 Linear Algebra Section 1.5 Solution Sets of Linear Systems Homogeneous Linear Systems: A system of linear equations is said to be homogeneous if it can be written in the form Ax = 0. This system always has at least one solution: FACT: The homogeneous equation Ax = 0 has a nontrivial solution if and only if the equation has at least one free variable. Example: Determine if the following homogeneous system has a nontrivial solution: x + x 4x = 0 1 2 3 2x + x x = 0 1 2 3 3x + 2x 5x = 0 1 2 3 1
Example: Solve the following homogeneous system: x + 2x 2x = 0 1 2 3 2x + 4x 4x = 0 1 2 3 2
Example: Determine if the following homogeneous system has a nontrivial solution: x + x x = 0 x 1 2 3 + 2 x = 0 1 2 x 5x = 0 2 3 3
4 Example: Let 1 6 0 0 8 1 0 0 1 0 2 1 0 0 0 1 5 1 0 0 0 0 0 1 A =. Describe the solution set of 0 Ax =.
Solutions of nonhomogeneous systems Ax = b Example: Describe the solutions of Ax = b where 1 0 2 A 1 1 0 = 2 0 4 and 5 b = 4. 10 5
Example: Describe the solution set for the system: x + x 4x = 3 1 2 3 2x + x x = 11 1 2 3 3x + 2x 5x = 14 1 2 3 6
Theorem: Suppose the equation Ax = b is consistent for some given b and let p be a particular solution. Then, the solution set of Ax = b is the set of all vectors of the form: w = p + v h where v h is any solution of the system Ax = 0. 7
Example: Describe the complete solution set of the system: x + x 4x = 4 1 2 3 2x + x x = 20 1 2 3 3x + 2x 5x = 24 1 2 3 8
Example: Let 1 6 0 0 8 1 0 0 1 0 2 1 A = 0 0 0 1 5 1. Describe the solution set of 0 0 0 0 0 1 5 9 Ax =. 4 2 9
MATH 2331 Linear Algebra Section 1.7 Linear Independence Definition: A set of vectors { 1, 2,..., p} the vector equation v v v in R n is said to be linearly independent if c v c v c v 1 1 + 2 2 +... + p p = 0 has only the TRIVIAL solution. The set is said to be linearly dependent if there exists weights c 1, c 2,..., c p, not all zero, such that c1v 1 + c2v2 +... + c pv p = 0. In other words: The set { 1, 2,..., p} v v v in R n is linearly dependent if one of the vectors in the set is a linear combination of the other vectors (not including itself). 1
1 Example: Let v1 = 2 3 Is the set {,, } 1 2 3 relation among them., v2 1 4 = 1 and v3 = 2 1 12 v v v linearly independent? If dependent, find a linear dependence 2
FACTS: { 1, 2,..., p} v v v is linearly independent if and only if c1v 1 + c2v2 +... + c pv p = 0 all ci = 0. If one of the vectors in the set is 0, then the set is linearly dependent. If there is only one non-zero vector in the set, the set is linearly independent. If there are two vectors: { u, v } is linearly dependent if and only if they are parallel (that is, one vector is a multiple of the other). Otherwise, L.I. 3
If there are three vectors: { u, v, w } is linearly dependent if and only if they lie in the same PLANE. Otherwise, L.I. Theorem: Any set { 1, 2,..., p} v v v in R n is linearly dependent if p > n. Example: Linearly dependent or independent? S 1 1 0 2 2, 1, 1, 1 = 3 0 1 1 S 1 1 0 2, 1, 0 = 3 0 0 4
S 1 5 2 10 =, 3 15 1 5 S 1 0 2 1 =, 3 0 1 2 S 1 1 2 =,, 2 1 1 S 1 1 =, 2 1 S 1 1 1 2, 1, 0 = 0 0 1 5
6 Important fact: Let 1 2... n A a a a = be a matrix. The columns of A are linearly independent if and only if the equation 0 Ax = has only the trivial solution. Example: Are the columns of A linearly independent? 1 2 1 1 1 0 2 5 3 A = 1 0 2 1 1 4 2 0 5 A =
Question: 2 u R, Span{ u } = 2 u, v R, Span{ u, v } = 3 u, v R, Span{ u, v } = 3 u, v, w R, Span{ u, v, w } = FACT: n linearly independent vectors in R n span the whole space. Extra: Let 1 2 3 8 E = 1, 1, 0, 7 1 1 2 3. What is the span of this set? 7
MATH 2331 Linear Algebra Section 1.8 Linear Transformations A transformation (or mapping, or function) from R n to R m is a rule that assigns to each vector n T :R R x R n a vector T( x) R m. m Domain of T: R n Codomain of T: R m For x R n, T( x) R m is called the image of x. Range of T: the set of all images - { T( x) : x R n } Definition: A transformation T is linear if: (i) T( u + v) = T( u) + T( v), for all u, v in the domain of T. (i) T ( cu) = ct( u), for all scalars c and for all u in the domain of T. In other words, T is linear if T( cu + dv) = ct( u) + dt( v) for all scalars c and d, and for all vectors u, v in the domain of T. 1
Fact: If T is a linear transformation, then T ( 0) = 0. Reason: Fact: These conditions imply: ( ) T c v + c v +... + c v = c T( v ) + c T( v ) +... + c T( v ) 1 1 2 2 n n 1 1 2 2 n n Example: Let 2 2 T :R R be defined as x x + x T x 1 1 2 = 2 0 Is T linear? Justify your answer. 2
Example: Let 2 2 T :R R be defined as x1 x1 + 1 T = x 2 x2 Is T linear? Justify your answer. 3
Example: Let 4 3 T :R R be defined as x1 5x1 x2 T = x 2x x 3 0 x 4 2 4 Is T linear? Justify your answer. 4
n m Example: Let A be an m n matrix and T : R R be defined as T u = Au ( ) Is T linear? Justify your answer. 5
Example: Let A be an m n matrix and b R m n m. Define T :R R as T u = Au + b ( ) Is T linear? Justify your answer. 6
Matrix Transformations Fact: If A is an m n matrix then the induced transformation T :R n R T u = Au is linear. Notation: x Ax that is defined as ( ) m A: m n (m rows and n columns); domain: R n ; images are in R m - the images are linear combinations of the columns of A. Example: Let 1 2 A 2 3 = 5 1, 1 u = 4, 1 b 5 = 32, 8 c 14 = 12 Define 2 3 T :R R as T( x) = Ax. (i) x1 T = x 2 (ii) T( u ) = 7
(iii) Is b in the range of T? Can you find a vector x in the domain of T so that T ( x) = b? Is b the image of a unique x? (iv) Is c in the range of T? Can you find a vector x in the domain of T so that T( x) = c? 8
Example: Let 1 0 0 A 0 1 0 = 0 0 0, Define 3 3 T :R R as T( x) = Ax. x1 T x2 = x 3 Example: Let 0 1 A = 1 0, Define 2 2 T :R R as T( x) = Ax. x1 T = x 2 9
(Extra) Example: Let such that T ( u) 6 = 2 4 u = 1 and 2 v = 5. 2 2 T :R R is a linear transformation and T ( v) 12 = 8. T( 5u ) = T ( 2u 5v) = 1 (Extra) Example: Let e 1 = 0 and 0 e2 = 1. 2 2 T :R R is a linear transformation such that T ( e ) 5 T =? 4 1 4 = 2 and T ( e ) 2 2 = 1. x1 T =? x 2 10
1 (*Extra) Example: Let v1 = 5 and 2 v2 = 4. 2 2 T :R R is a linear transformation such that x 1 T = x1v1 + x2v 2 x. 2 Can you find a matrix A such that T( x) = Ax? 11
MATH 2331 Linear Algebra Section 1.9 The Matrix of a Linear Transformation Theorem 10: Let unique matrix A such that T( x) n m T :R R be a linear transformation. Then, there exists a = Ax for all x 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 jth column of the identity matrix in R n. ( ( ) ( )) A = T e T e 1 n This matrix is called the standard matrix for the linear transformation T. 1
Example: Find the standard matrix for the dilation transformation T( x) = 5x in 2 R. Example: Let 2 2 T :R R be the linear transformation that rotates each vector in 2 R about the origin through an angle θ (counter clockwise). Find the standard matrix. 2
Example: Let 2 R through the x1- axis. Find the standard matrix. 2 2 T :R R be the linear transformation that reflects each vector in 3
Example: Let 2 R onto the x1- axis. Find the standard matrix. 2 2 T :R R be the linear transformation that projects each vector in Homework: Study all transformations on Page 73, 74 and 75! 4
Definition: A mapping n m T :R R is said to be ONTO if each R m b is the image of at least one R n x. Definition: A mapping n m T :R R is said to be ONE-TO-ONE if each R m b is the image of at most one R n x. That is: T is onto if T ( x) = b has a solution for any b R m. T is one-to-one if for each b R m, T ( x) = b has either a unique solution or none. 5
Example: Let 2 2 T :R R be defined as x x + x T x 1 1 2 = 2 0. Is T one-to-one? Is T onto? 6
Example: Let T be a linear transformation whose standard matrix is: 1 2 7 1 A 0 4 5 1 = 0 0 0 1 Is T one-to-one? Is T onto? n m Theorem 11: Let T : R R be a linear transformation. Then, T is one-to-one if and only if T( x ) = 0 has only the trivial solution. 7
Theorem: Let matrix. Then, n m T :R R be a linear transformation and A be the standard (i) T maps R n ONTO R m if and only if the columns of A span R m. (ii) T is one-to-one if and only if the columns of A are linearly independent. 8
Example: Let 2x1 + x2 x1 T = 4x1 5x 2 x 2 x1 + 6x 2 Show that T is one-to-one. Is T onto? 9
(Extra) Example: Let 2 2 T :R R be the linear transformation that first reflects points through the x1 axis and then rotates points 0 90. Find the standard matrix. 10